Tere 


+ Oe ee eek “* ol caaliniad ue ~ 

~ = *. . e608 666 O84 “4ee% o yew Ce ma 

. *“" © e+ oh 2 2S wt ie - ~<e@ 

7 7 . - + ee oe - ee he ee eo 

“ws ° o* ose» Pe eo ee ee Ce 

+e ** a ee ee ee 1 > rere * Ke bA O45 | ee we 

a “Pes ee - ~* -) “ie er eee ee ee oe \ Puateen’ 

eae *- i4+¢ened a. a ee ee ee ee ee 2. eee eo | 

‘a i re ‘ve ee ee re er ee ee ee ee ee 

. ied ¢*erse @eevpe . - r= ee ~~e*e ee ew . 

t~-~ & be , 9 

. ‘ . » . er ee i Ans Arts Prey 
« 264 eee ey Oe we me me eR ate Op mee ope eee tok A eee on . 
“> Se » » 1 ED th OO ee me Oe mir nw Py Trees wore ep Pa a 
‘ “4 s pip Paw ete ep ee ew Od dan & ‘ Certainty Pde Mo oe i oe 
a ewe ? ee ee ee ee ee ee ee ee oe ok See hie wk EP A Tet 

" ” ++ +a eseo sane Oe4 ein wr @ eee» BG my Cg ae a ty ee de ae et OIE poe ergy 
iiedete - i a a ee he eo ie en ee ee ee hee ee ee eee ae oe | 4 
Se PMO Feb hee e os ee LE ee ee ee bet Pe ONY & Se Ee REE Oe dee oe 8 yah Aare teehee aA ‘ 
a ale A ee oe OT he ee eee Pa ee & eae rrr reek ey Pee hPa Bw é 
OPA oye pte eae es ee Perr ees ee es paper h chu a bX : 


reo By Oe eb A 
ee ee od 


Suk 


Pe oe ee . j 
Oe eo ee Oe eed 
eee ee ree er ee woe ey 
het Ak ee Oh he me pee ee ge ple ee Ose @ ON a oe 6 ~ 
Lem Gum © Pee DD hee le dt tela Bee ate > pe tee le we i pied oe ey be 
Pee ee ee ee ee ee ee ee ty 


~Pe rp da- 


aH 


4 > © by 
pee eens 


= hme eb 


“Tete 
oe 
= 
= 


Pet ee ee ER aryl itis le te ey Dh te hm eon sto ap ge de 
Ok he Fd HO FE at de ge © Rte ESE SET Fe Gh hi ea ri Pom Ba i ¥ d 
Nee ee en en eee ee ee ee eee ee Ce ee aor LY 6) 19 Sb Pa eA aes 

ee eee ee Re eee ee ee ee ee WK ae ae tte te he pe Rh ge dy 
See eh eee Ee PE ae gee med A EE ee OA Te & PR a he re 7 
ee ee ee ee ey bd soll Pulido Mie etal. | 
«+4 em om ye Patt Fade deukhd 


eek Oe et Ae be ha 
aah es epee te the 


oth @ © : 
jveulee lane ’ “4% : = Be ah Gt A, Fee ete 


~~ ’ sk 


-* 
ot ieee oe eo 


Wim Hee me 


4 
WA Ae eee ee hee 


. ; , yhr tt 
Se ee. St ee ee Se te ee ee ee ee ee ee ae ek 
, ’ = hee ee OER Pe ee oe Pe hee Fw & me Gwe 88 iy Me San rd aE op ey : lee 
i & wad Oe we de) a ee ee a ee ’ 


u% ‘a4 o> & Od 4) S & ee ee ee eo em ee ee ee 
ee er a re ee 2 ee a Se oh Haye 


Pelee Me ee ee oe oe Vy © oe 
ee ee a Be eee ye RT Behm teem Ge OF Ie 
Oe ee ee ee ee ee eee ee 
err a ree Soeete eae Wee ee ee ee a ees a ee 
Ah) bow eee rw er - 
ee oe ee ee 
‘es Sear es 


a 


Cy 


has 

& . 
Bd a ip 7 
Fats agp Natiete Ae 

; Pas he aces A oe 


Paty 
Pusey ee cw 


7 
> 
¥. 
¥ 
?. 


? 
‘= 


“yt 
os 
> 
*. 


ey 
2 

= 

»* 

4 
it 
: 

= 

at 

~ 

fal 


€, 
z 
ri 


ete © some ee et eee ee eee sod 2 Pa ie 4 
ee PER Ee PEE H bl Oh fe AR Oe eb ef ar hee 2 i & ir opt Le ae ey 
OO eth i me ee oe PB dee dine Horie AP lip pay ee ait ik Fo kee eo 


eee ee re ee ee a ee ee 4 
a ee ee ee ee een ee tome Vet a oe ee ee a ees 
YS 56 ey et eit 

+i ew amr he 
forge nia a 
+e fe ey 


> re he 
: ee 
gh  % pete 


74 


a 
< 
7 
>. 
Cz 
e 
2 
z. 
25 
4 


& 

+ 

= 
— 
- 


ee ee ee ee 
Sn a on ee ee ee 

Pensa we wade ae * F&F 
aap te pe POR tee ge 


+. 


+ ' 
4 


sok: Se en ne ee ee ee te ee ne ee aoe ar oe ee oe ee ae ke ‘be 1 Lhe iae Me mak 
ae ee Sd ee et eee ene ee eo oe ee ee ee oe oe) nthe : tye 
et ee ee ee ee ee ee ee Hy Pact gt lag oP th Diels Maly ae Sh 
ee ee ee ee ee ee ie oe pe ee pep die theron Spoken, (Deak abe Hemet 
See ee ea ee ee rer ae pean ep ey birst 4b G44. ws elie he 


he (hy 


Pi ee ee me 
OS i ee te Pee a a ng ae eee 
re ee en ee 2 oe 


hn Se 4 es & pe Be dg ee iene  & Gee Oe ‘ o mikbbay ewes 
7 SE be ig ee ha re eed S fm ctor > ert wm bw 
» a few on et « . Pare es on re 4 we Se GOS He a Oh meine" 


“ : 

oe See eo a ae oe reer es ete Pen ome ee ee ear me ee 
Oe ee ee ee ee ee ee ee et 
> hie eb ee ew tek whe eo 
‘ ey ORS a SP atlg 

be Ss Jud ew ne 
hie ED 

_ 4 here Ot 
Mee ROT oA epee ge et 
tae heen a 
has lp be nd AA ae : 
Ce ee ee ee we ere ee ae ok he 
oe eee eee ee ee ee ee ee ee 
Pe Ri ee ty ts ae oe Dei oti Me I BeBe ttle dah. wri Fal te te Way 
+P te te Men yp te tea! bed & ey te Bek wow 
Rebtel ee ee lew 
he ® Hy ee 


Me. 
3 


he 
Tt 
gg 
Se) 


2» 
24 
2% 
2 


a 
= 
= 
*¥. 


. 
‘< 
¥. 
ad 


* 
4 
x4 
es 
> 


teat wa ee ry eta 
ant @ rat ead Hei en ihe Beh re Ay de ne tog ee ’ ry 
2 His hee we ber A Ba hud Bit; et 4: pore 


4 tot , Y 
© bey fe tos te Ry 1 Bee yi Repo ye we ra i eh 
are v oe & i eetee Loe ee A Sb eA OF yan 
eee ea mba ed ned erpie Ahi oe ee ® oe a ee ee ee ee eid 
thee welt Pe ee ee. ee ee a ee ee ee er ee | 
mse we ' he oj tr bk el OS Re Glad d geared 44 he BE ee De A aN ay oe ae oe FM Boley Site shite 
Bei eit te me ee hey mm i deee Wh te hie he Ale Tok of ae tee thtcd al dere fat) Fah he ye ie peered ay + age gh et 
wad te he he ek aad ep wy ei od ne ae RB Ge a OG oh Now Te dR owe grat More (ed 
pts + ee a ee ms eer ol Sit Ree he & ) 
-ee a ee ee ee ec re ke eT eee ke 
+< 4 ee a4 Ne ee ee as a 


ee oe oe a fae , . 
oe ee ee ee er ee ee Burov 
Po ee ee ee ee ee ee ee te ka! im ti 
. - had eS ee et ee ee ee ee, ry S f 
teiiiday 4) te a we Cre 2 ae wr See eet nae ee i em a Me aegis 
tbat To hr ee ee ee oe a Hee he 
2h EN ete Bran a ee i Oe po Ge eg lige Dare hy bedi le eth Dae bah 


rr. 
3 


ee ee ee ee ee ee es ee ee a VERE Meeps hs ge be 
a ae ee oe eee en ee ee eo ee eg i 


tei ie ts ee 


yt 
cae) 


rf 
a 
5 

sfete 
epee 
ae 


~ + © ss ' ee ee ee ee eo mere ee ay 
wy st ‘« tree erg ee Rk ee Ae Beet Be unite ete age gethe Sig a th las oo 
a ar We bel he hime Sede ot ie ee SS ER 
. rr Ae Tae the Weide Syme ge he acl apm he ete inch ny oe eat Tee She heh Rei i ae: 
te font at A TR ey Coe he ee Metre he dei age im Nerd rad tots Sethe hh We Ny Ae epee ois de Bete 
Ce 7 el tate ies, Vode Ait eat tte et et ok rk ee we Wee ae ot) Wwe ITs Yeet) a tthe dt 
6 ae ee ee ee a i ek Str Gee Gere me Were eg aetbed oie Ver ed be ipel 3 eo La 
we oi Pw Be a fee me ef net) ae eee ee Ryan he he tetee th Petgsdpard & te he de ye 
Peo He BH ardee o ee ee ee ee or ee eee ee : Sup de 
ap &® . 4 Oh Re re bee ere Pe ee eee ee, eee oro * s 
2 eo ee ee oe erent a eT re eee es ae ee eee 


@tdin 


= 
* 


2 ee ee ee en ee ee a ee ee ee 
et ke ed ee a a ra ot eee Sy Ser ny re 

he hele wb Te a atk, aid a DO. ne as ee 
Aah eM eH tee 8 ed wed ite 
Pep See Se ee ee 
Pam Hed eS ao eh mee MH hee thee 
be heeded GQ tote te BP the th A ®. eip ho B 
heh wal & te > i, ee ea cee ae 


atest 


a 


ars 
+ 


Am ee we ew bee r 


me Re A eae he be fad de 


ey eebeth, S Bete be bee, ee ’ f f 
ee ote he “he hye we ee 64 ke He ie rena tae the the Aa eb ira ob ie rds URN Byte hog af eM og mee Hy 
a ES rl & eo lee) Oe Ae.o ab Oe ak es Be os 0 le A eh Gidea Gh ie wie aol woe ach obec ila tees iy 
Av he Qe Bee ae donde Fee. & & Se . RE Re He ee RR nee & nD eek do habe dye a i 

et ew fe BD nh Pekin atime derngs oe | ah Gel we Gedee ae Ce 
-~4 “s omiete dob a big wie Fed ee os 
e- ae * 4 > poh oe eo ee er 
» + = ‘ ru * , 7) > tb smh id eS eighty y i: ike be ot 
~=* ee eeeen ee he ee te get he ee 
<= he ee dt & ee Eh we oo» 4 oS paA AY 
—. ee > we > he ren a? i aw» ae eee ‘ 
on« a em re be aia ho Odea, ot Oe Raed mde le 4 
COR eee Rh ite we ey kad . js 
~~ ee wh He eee bee ee be Pde ew * 9 
~a. te wi dt ea Se oe ee oe) de® . 
Ww £9 4 eal e a aa me oO ok ee He 
the ere rAd Gawd bee! ey 
Serer rs se ae 2 Varese a, oe cata 
“ ee ee) Raw AA ee hee Ey te © Me Be teehee sea 
ee oe | "ee Se ee tow Me yS: 
rode bir en we nw & tos hot gt a he eter ay * wee al ke i 
+ 4 |e Oe ea 4 & & re eee 4 7 
ee ee ed ee ee ees ee) ‘ 
Te ee tee Re Oe ee oH we Oe 54 4 
_- - = - oh ei) & te & hint dw a ee) * 
. ne a 2. *“- sore thee > APE PA peed es ee Ye ak 
Di onde in hah. @ i arte haan & a a oe Co en ee ee eo oe ee ee cee ce ae ae 
“ : . 44 b BP hate Oye wh ee Rw tnlh dye ee 


oe me A rh hp rat < We ‘ 
We ok, eee uated. . 
J enn “i ta eatery sation 
te WN Bo hd iedyy 
cay » dow » ee iy ete deal 
ims a 


ar} 
es a Ae Sr etal ah ara » 
Sb ae ete hy ty a a eS 
ie te Oe Rett de toed dete 


adh & As lente tek 9 ba es tae 


ie ts kote OO a le we Bae he cused Sine 
eh SUR 


oe a ee es 
te ee eo Se ee a 
ee aa Ae 8 te Per ew em 


eee ee ee eo a ee ee oe ee er ee ee Pee en hr vy are wey Sota tee 

Fe ee tee hehe oe be ay Moet oe tn AEH fetes ae Sek be dete d db tebe oy wae We EO bere denier 

ot) Ae etm te a we ee eRe Ore A " 

ee | 
- 


aS soe ee 
ee ee 

white wk Ee Arh 
‘ Fee? ASTerre Wow Oa Ss 


Pathe eee de he oy tee ace rb 
bebe Sed be be ati ot he hel theme 
toe ae ek he 


sie’ 
oe = btw 


Math. 


The person charging this material is re- 
sponsible for its return to the library from 
which it was withdrawn on or before the 
Latest Date stamped below. 


Theft, mutilation, and underlining of books are reasons 
for disciplinary action and may result in dismissal from 
the University. 

To renew call Telephone Center, 333-8400 


UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 


AUG, 24 198 
JUL 34 RECD 


L161—O-1096 


Digitized by the Internet Archive 
in 2022 with funding from 
University of Illinois Urbana-Champaign 


https://archive.org/details/treatiseonisopei00wood 


t-. TREATISE 


- 
x 


ISOPERIMETRICAL PROBLEMS, 


CALCULUS OF VARIATIONS. 


a 


‘BY ROBERT WOODHOUSE, A.M.ERS. 


FELLOW: OF CAIUS COLLEGE, CAMBRIDGE, 


CAMBRIDGE: 
f Printed by J. Smith, Printer to the University ; 


AND SOLD BY DEIGHTON, CAMBRIDGE; BLACK, PARRY, AND KINGSBURY, 
i LEADENHALL STREET; AND MAWMAN, POULTRY, LONDON, 


1810, 


fea <: 


: ey SP ‘ eye 4s 
OF AER TE Came 4 Pte GA BE ae, 
: A Sar ha ee 


ae, 
deo hererpdyinty Mp - 
‘ ‘ eanee 
rd i Li 

tes 


sy OE 
* at 


4.56 


TF IR 


PREFACE. 


‘Turret needs, perhaps, no other apology 
for the present Work, than the mere statement 
of the fact, that there is, on the same subject, 
no English, and only one Foreign Treatise *, of 
which the celebrated Euler is the Author. 


The copies of this last, a work of uncommon 
merit, are in this country very rare. ut, not- 


_ withstanding its various excellencies, a mere 


translation of it would not have rendered unne- 
cessary the present treatise, since, independently 
of any other consideration, several of Euler's 
most important processes are, by more concise 
ones, now Superseded. 


But although there is only one separate Trea- 
tise, the subject itself has been discussed in 
several analytical works, English as well as 


* Methodus inveniendi Lineas curvas proprietate maximi 
minimive gaudentes, Lausanne, 1744, 
1. 
v 


206208 


b | PREFACE. 


Foreign. Our countrymen are accustomed to 
assign, in their treatises on fluxions, a chapter to 
the maxima and minima of curves; and, the 
foreign mathematicians consider the same subject 
under the head of ** Calcul des Variations.” 


Maclaurin, Simpson, Emerson, especially 
the two latter, have not explained the principles 
of the subject with sufficient perspicuity and 
precision; and, in point of depth of research, 
there are various problems, to the solution of 
which, their formule are totally inadequate. 
Indeed, which is remarkable, there is, I believe, 
no English treatise that mentions Euler's 
formula of solution; which, for the simplicity 
ef its construction, and facility of application, is 
exceeded by none in the whole compass of 
analytical science. 


The foreign mathematicians, Euler, Le Seur, 
Bossut, and Lacroix, have furnished their treatises 
on the differential and integral calculi, with 
formule of solution adequate to any case that can 
be proposed. ‘There is no deficiency on that 
head. But, these Authors too suddenly carry 
the reader into the middle of the business and 
immerse him in calculations; and, if they soon 
provide him with the instruments of solution, 


PREFACE. ill 


they instruct him neither in the object nor the 
principle of their construction. 


This is not said in the spirit of an Author 
preparing, by the censure of preceding Works, a 
reception for his own. For the truth of the 
statement, I would appeal to the experience of 
all who have consulted the above cited Authors ; 
besides, it is not difficult to assign the reason 
of the alledged defects. 


When Lagrange in 1760, published his new 
method of solving problems of maxima and 
minima, he composed his memoir for mathema- 
ticians, familiar with its subject, and well versed 
in the researches of the Bernoullis and of Euler. 
Accordingly, he very briefly states the principles 
of his calculus, and enters into no explanation 
of the nature of the subject. His compendious 
method of computation, however, has been 
adopted ; and subsequent Authors have com- 
posed their treatises very much on the plan of 
Lagrange’s memoir, with some, but slight and 
imperfect, preliminary explanation. These 
Treatises, however, the student is expected to 
understand ; that is, if the matter be fairly stated, 
he is expected to understand an intricate sub- 
ject, with advantages much less than consum~ 


iV PREFACE. 


mate mathematicians before him enjoyed; since, 
there is neither proper explanations presented to 
him, nor is he directed, by way of preparation, 
previously to consult the Works of Euler and 
the Bernoullis. 


Such are, in my opinion, the defects of ex- 
isting methods ; still, however, I have not com- 
posed a treatise on the subject, by merely 
remedying them; that is, by inserting formule 
of sufficient extent, and by more fully explain- 
ing and illustrating their principles. But, on 
a novel plan, I have combined the historical 
progress with the scientific developement of the 
subject; and endeavoured to lay down and 
inculcate the principles of the Calculus, whilst 
I traced its gradual and successive improvements. 


If this has been effected, which I think it 
has, in a compass not very wide of that which 
a strictly scientific treatise would have required, 
the only serious objection against the present plan 
is, in part, obviated. For, there is little doubt, 
the student's curiosity and attention will be more 
excited and sustained, when he finds history 
blended with science, and the demonstration of 
formule accompanied with the object and the 
causes of their invention, than by a mere ana- 
lytical exposition of the principles of the subject, 


PREFACE. Vv 


The plan, perhaps, would not suit any other 
department of science, so well as it does this ; 
which is limited in its extent, and has had but 
few, although eminent cultivators. 


Other advantages, besides that of an excited 
attention, may accrue to the student from the 
present plan. He will have an opportunity of 
observing how a calculus, from simple beginnings, 
by easy steps, and seemingly the slightest im- 
“provements, is advanced to perfection; his 
curiosity too, may be stimulated to an exami- 
nation of the works of the contemporaries of 
Newton; works once read and celebrated: 
yet the writings of the Bernoullis are not anti- 
quated from loss of beauty, nor deserve neglect, 
either for obscurity, or clumsiness of calculation, 
or shallowness of research. ‘Their processes 
indeed are occasionally somewhat long, and 
want the trim form of modern solution. ‘They 
are not, however, therefore the less adapted to 
the student, who is solicitous for just and full 
views of science, rather than for neat novelties 
and mere store of results. Indeed, the Authors 
who write near the beginnings of science, are, 
in general the most instructive: they take the 
reader more along with them, shew him the 
real difficulties, and, which is a main point, teach 


v1 PREFACE. 


him the subject, the way by which they them- 
selves learned it. 


In a former Work *, I adopted the foreign 
notation, and the present occasion furnishes 
some proof of the propriety of that adoption. 
In the Calculus of Variations, it is necessary to 
have symbols denoting operations, similar to 
those that take place in the Differential Calculus : 
now, d being the symbol for the latter, d isa 
most convenient one for the former: analogous 
to 6 there is no symbol in the English system of 
notation. If then I had used the fluxionary 
notation with points or dots, I must have in- 
vented symbols corresponding to 6 and the 
characters formed by means of it. But, the — 
invention of merely new symbols is in itself an 
evil. M. Lagrange indeed, whose power over 
symbols is so unbounded that the possession of 
it seems to have made him capricious, has treated 
the subject of Variations without the foreign 
notation; this he rejects altogether ; and, which 
is strange, has employed the English notation, 
but not adopted its signification. Thus, with 
him, # is not the fluxion, but the variation of zr: 
the fluxions or differentials of quantities are not 


* Principles of Analytical Calculation, 1803, 


PREFACE. Vil 


expressed by him, but solely the fluxionary or 
differential coefficients; thus, if «be a function 


of x, 2 (= a Sty = =) is the differential co- 


efficient. What advantages are to arise from 
these alterations it is not easy to perceive: yet 
they ought to be great, to balance the plain and 
palpable evils of a confusion in the signification 
of symbols, and of the invention of a system of 
notation to represent what already was repre- 
sented with sufficient precision. No authority 
can eyen sanction so capricious an innovation. 


There is another point towards which I am 
not unwilling to draw the attention of the 
reader; and that is, the method of demon- 
stration by geometrical figures. In the first 
solution of Isoperimetrical problems, the Ber- 
noullis use diagrams and their properties. Euler, 
in his early essays, does the same; then, as he 
improves the calculus he gets rid of construc- 
tions. In his Treatise*, he introduces geome- 
trical figures, but almost entirely, for the 
purpose of illustration: and finally, in the tenth 
volume of the Novi Comm. Petrop. as Lagrange 
had done in the Miscellanea Taurinensea, he 


* Methodus inveniendi, &c, 


Vill PREFACE. 


expounds the calculus, in its most refined state, 
entirely without the aid of diagrams and their 
properties. A similar history will belong to 
every other method of calculation, that has been 
advanced to any degree of perfection. 


The plan of the Work has been slightly de- 
scribed; and indeed it scarcely requires any ex- 
planation. On that, however, I chiefly rely, as 
the means of rendering easy a subject, which is 
acknowledged to be one of the most difficult. 
But, although I am not aware of having omitted. 
any thing that is requisite to the full explanation 
of the subject, yet I cannot flatter myself that it 
will be thoroughly understood from this Work 
alone. For, in general it may be laid down as 
true, that no doctrine, of novelty and intricacy, 
can be completely taught by a single Treatise. 
It seems to be indispensably necessary for the 
student, that the subject should be put under 
several points of view: that if not apprehended 
under one, it may be under another, For this 
reason, though not wanting an Author’s par- 
tiality for his own performance, I recommend 
the perusal of those works, to which frequent 
reference is made in the following pages. 


PREFACE. 1X 


I return my sincere thanks to the Syndics 
of the University Press, who, very liberally, 
have defrayed two-thirds of the expense of the 
present publication. 


— Caius College, 
_ Nov. 17, 1810. 


Tasie of the Foreten, and the corresponding 


Encuiso WNofation. 


F.| d xe | dx 


d*x dx? dx 


"3 
7 du | du | du | du au rr (4) 
dg | Ge ae | de.dy |" da dy dz 
& v3 " Ly vy AS 


List of Foreign Authors that treat of the Subject of the 
present Work. 


BERNOULLI, James and John. — Works. 
fComm. Acad. Petrop. tom. VI. VII. 1733, 
1736. 
Methodus Inveniendi Lineas curvas proprie- 
EULER. tate maximi minimiyve gaudentes, Lausanne, 
, Geneva, 1744. - 
Novi Comm. Petrop. tom. X. 1766. 
Calcul Integral, 1768. 


Miscellanea Taurinensia, tom. II. 1V. 
LAGRANGE. < Théorie des Fonctions Analytiques, 1797. 
Lecons sur le Calcul des Fonctions, 1806. 


U 


BorpA, et Fontaine.—Acad. des Sciences, Paris, 1767. 
Li SEur et JAcguier.—Elemens de Calcul Integral, 1768. 


Sauri.—Cours de Mathematiques, tom. V. 1774. 
LEGENDRE.—Acad. des Sciences, Paris, 1786. 


Lacro1x.—Calc. Diff. et Integral, 1797. 
Bossut.—Calc. Diff. et Integral, 1798. 


~ISOPERIMETRICAL PROBLEMS, 


AND THE 


CALCULUS of VARIATIONS. 


CHAP. I. 


The Problem of the Curve of Quickest Descent proposed. —~ James 
Bernoulli’s Solution of it.—Principle of that Solution. 


Tue ordinary questions of maxima and minima were 
amongst the first that engaged the attention of mathe- 
maticians at the time of the invention of the Differential 
or Fluxionary Calculus. This invention took place about 
the year 1684, three years before the memorable era 
of the publication of the Principia. But, the principles 
of the differential calculus, were not, like those of phy- 
sical astronomy, given to the world, at once, and as it 
were, on a sudden, in a formal treatise. They were 
communicated gradually and by piecemeal ; in letters 
between men of science, in small essays and tractates, in 
solutions of particular problems published chiefly in 

B 


to 


Newton's Problem of the Solid of least Resistance. 


a work intitled the Aeta Eruditorum edited at Leipsick * ; 
which, at that time, was the common vehicle of commu- 
nication between men of science and literature. There 
is scarcely a formula or process, which we now find 
compactly stated in our modern treatises, that was not ~ 
in those Acts the subject of some correspondence and 
discussion. The methods of drawing tangents, radii of 
curvature, of determining the points of inflexion, the 
maxima and minima of quantities, &c. were treated as 
subjects of great importance by Leibnitz, and by the 
two Bernoullis. 
e 

The first problem relative to a species of maxima and 
minima distinct from the ordinary, was proposed by © 
Newton in the Principia: it was, that of the solid of least 
resistance. But, the subject and doctrine became not 
matter of discussion and controversy, till John Bernoulli, 
Professor of mathematics at Groningen, required of ma- 
thematicians the determination of the curve of quickest 
descent. ‘This he did in the Letpsick Acts for June, 
1696 +, p. 269, under the following form: 


* Problems proposed and solved, became an amazing means of pro- 
~ moting the methods of calculation, or the calculus, to which they 

belonged. Of this John Bernoulli was aware, when, in his famous 
Programma relative to the line of quickest descent, he says, ‘ Cum 
compertum habeamus, vix quicquam esse quod magis excitet generosa 
ingenia, ad moliendum quod conducit augendis scientiis, quam 


difficilium pariter, et utilium questionum propositionem.” Opera, 
tom. I. p. 166. : 


+ Opera, tom. I. p. 155. 


John Bernoulli’s of the Curve of Quickest Descent. 3 
<“ProBLeEMA Novum, 
ad cujus Solutionem Mathematici invitantur. 


a *¢ Datis in plano verticali duobus punctis 4 et B, 
* assignare mobili M viam AMB, per quam gravitate 


** sua descendens, et moveri incipiens a puncto 4, bre- 
** yvissimo tempore perveniat ad alterum punctum B.” 


Six months, the time allotted for its solution, being 
elapsed, and no solution appearing, John Bernoulli at 
the request of “Leibnitz (who intimated that he had 
solved the problem) prorogued the term, and again 
anounced the problem in a programma* edited at Gro- 
ningen in January 1697. In the May following, his 
brother, James Bernoulli, professor of mathematics at 
Basle, published its solution in the Acta Erud. Lips. 
p- 211+; and after the following manner: 


* Opera, tom. I. p. 166. 
+ See also his Works, tom. II. p. 768. ; 


4 James Bernoulli’s Solution. 


Let OGD be the curve; conceive a portion of it CGD, 
to be divided into two parts CG, GD; and take another 


O if 


may “ 


element to the curve CLD, divided also into two parts 
CL, LD, and indefinitely near to CGD: then since by 
hypothesis, the time through CG+GD, is to be a 
minimum, and since quantities at or near their state 
of minimum may be considered constant (for their 
increments or decrements are very small), we have 
t.CG+t.GD=t.CL+t.LD 
[¢ . CG, abridgedly representing the time through CG. | 
and »» ¢.CG—-t.CL=t.LD—t.GD 
agan CE:CG::¢t.CE:t.CG 
[ CG considered as an inclined plane. | 
and CE’ CL tek: t. CL 
consequently, CH : CG— CL [| MG]::t.CE:t.CG—t.CL 
but MG : LG : EG: CG 
[by similar As LMG, GCE. | 
». CE: LG:: EG x (t.CE): CG x (t.CG-— t. CL) 


similarly EF: LG :: GIx (¢. EF):GD x (¢. LD—t.GD.) 


. Curve of Quickest Descent a Cycloid. 5 


Hence, equating the two values of LG, we have 
EG x t.CEx EFx GD=GIxt.EFxCEx CG 
, CG GD 


“ EGx Tae” EFx GD= CIx ite * Ex CG 
. [substituting for ¢. CE, t. EF. | 
EG GI 
CHE’ WHE * 2 CE: EF: CG.:GD,[1.] 


Be, is a property of ihe cycloid *: hence the curve of 


quickest descent, or the brachystochrone, is a cycloid. 


> 
> 


_ The proportion [1] may be thus expressed : 
EG. GI 
GC‘ GD 

sin. 2. ECG:sin.L GDI:: (/HC: /HE :: vel’. at C: 


vel’. at G, or the sine of the angle made by a vertical and 


: YAC: SHE, or 


an element of the curve is proportional to the velocity. + 


* This will easily appear by constructing the figure with its 
generating circle ; or, the known equation of the cycloid may be thus 


deduced: since a ees we have BoA Rise eee 
eae /ACVCGT 4H. OD Vy.dz vy .dz 


and since da’, 7’, dz’, are the consecutive values of dx, y, dz, it 
y 


_ follows from the above equation, that 


id de 
is every where the same, 
Vy. dz : 


: ] 
or is constant, and therefore may be put = Fe" Hence, ad# = 


y.d?+y.dyanddr= dy, the equation to a cycloid. 


Vy 
v(a—y) 

t Lagrange, Cal. des Fonct. 8vo. p. 425, says that this was the form 
of the result obtained by Bernoulli. It is not however the form given 
in the first solution that occurs in Bernoulli’s Works, 


6 Principle of Bernoullt’s Solution. 


In the preceding demonstration two principles are in- 
volved : one* borrowed from the doctrine of the ordinary 
maxima and minima of quantities: the other, at the time 


of the first discussion of these questions entirely new, and 


in fact assuming ; That the property appertaining to the 
whole curve, belongs also to any element of the curve : for 
of the preceding problem, the condition was, that the time 
down the whole curve OCD should be a minimum, 
that is, less than down any other curve, passing through 
the points O, D: but in the demonstration, an element 
CGD is taken, and it is assumed that the time down 
this element is also a minimum: so that, the whole 
curve is determined, by determining a portion or element 
of it. The principle undoubtedly is true, both in the 
preceding and in other problems, but it is not univer- 
sally true. Of the exceptions, and of their reasons, 
we shall hereafter speak, and here only briefly notice, 
that the Bernoullis and Brook Taylor, preface their small 
tracts on this subject, with this principle, establishing the 
same nearly after the following manner : 


If the curve APB (see fig. p.3.) has a certain pro- 
perty of minimum, a portion of it PQp, has the same 
property : for suppose 4P, Bp, to be determined, then 
if PQp possesses not the property of maximum or mi- 
nimum, suppose another small portion Pgp to possess 
it; therefore Pgp added to AP, and Bp, will form a 


curve, that in the case of maximum will contain more, 


* See page 4. line 5, &c. 


: 


Explanation of the Principle. 4 


or of minimum, less than APQpB; which is contrary _ 
to the hypothesis *, ia 
? he first principle is, as it has been remarked, that 
Wire is employed in the ordinary questions of maxima 
and minima. If, for instance, we wish to determine 
when anyordinate y is a minimum, dy its differential is 
put equal to nothing. Hence, the contiguous or con- 
“secutive ordinate which generally is, 


4 
Y=q: pe T 5 


def « ; 
fe (Ax) + &e.t 
is in this case reduced to 
dy y)re 
Seen? Oe) 


which, if we make Av infinitely small, will differ from 
the ordinate y only by infinitely small quantities of the 
second order; and if we neglect them, may be said to be 
equal to the ordinate y. Of this nature, and requiring 
this explanation, is the equality in James Bernoulli’s 
demonstration, when the time through CG + the time 
through GD =the time through CL + the time 
through LD. 


* P. 296. tom. II. John Bern. Leip. Act. 1701, p. 213. James © 
Bernoulli, the author of the essay, as if suspicious of the truth of his 
principle, says, ‘‘ Sensus theorematis vel demonstrationis ejus videtur 
paulo obscurior, nec satis determinatus: sed planior et infra ex appli- 
catione, &c.” John Bernoullialso, Acad. des Sciences, 1706, p. 236. 
lays down the same principle: ‘‘ Parceque toute courbe qui doit 
donner un maximum, conserve aussi dans toutes ses parties les loix de 
ce méme maximum, &c.”’ 


+ Woodhouse, Prin. Analy. Calc. pp. 44, 163. 


8 Object of the common Doctrine of Maxima and Minima. 


The use of the principle, however, in Bernoulli’s pro- 
blem, and its illustration, will enable us to point out 
the real and essential distinction between the ordinary 
calculus of maxima and minima, and that which is the 
subject of the present treatise. In the former the relatio "ae 
of y to x is supposed to be given, or, in other words, the 
function f in the equat*. y= fa is supposed to he known ; 


and the process of equating y and y + zo Ax+&c., or 


of making dy =0, enables us to find a particular value 
of x which substituted in y=fx gives the greatest or 
least value of y. Thus in the equation y= (2axr— 2° 
+h), fa = /(2ax—2°+5) and if the greatest ordinate 


Jp 
p 
oo 4A 


be required, (puttng MQ=y, and MN=Ac), NR=y+ 


dy LY 5 ei dy_a-«x | 
a Ar. + rere (Ax*)+&c. and—~ = = 0, conse- 


quently x=a, and the maximum value of y= +/ (a’ +). 
In the latter calculus, on the contrary, the relation of y 
to wis not given but sought, or in other words the form 
of the function fa [that is, whether it is /(2ar—2*) 


‘7 f } 3 Md : 7 } 1 
or €™ or (ax” ~ ba)", &c.] is the object of investigation. 


Object of the Doctrine of the Maxima and Minima of Curves. 9 


For oy reason, the augmented value of y=1/Q is not, 
as before, its consecutive value NR=2 yta y, but Mq = 
y+@Qq=y+», which must be substituted for y in 
— any function of y. ‘Thus, if the equation 


etween y and x, or in other words, if the function fx 


is required, such that of (ax — y’) y.d«x shall be a 
_ maximum, we have, changing y into y +», the consecutive 
value of the expression for the maximum, (axy — y’) 
+ (ax—3y’)v+ &c. and since this expression must be 


di 
a maximum, the second term, analogous to ~ Avx in the 
: ax 
former instance, must = 0, or av —3 y’ =0, andy = Nhs 


Here it is plain, the result gives the form of the 
equation, that must subsist between y and «, so that 
generalizing, it may be said, that it is the form of the 
function fx in y=fx, which is the object of search. 
The two preceding instances plainly shew the points of 
agreement and disagreement of the two calculi. 


Before we quit this part of our subject we will 
shew how Bernoulli’s method of solution, without any 
alteration of principle, may be abridged. Since the 
time varies as the space divided by the velocity, and 
the velocity as the square root of the height, we have, 


oe rim CG GD 

SHC’ SHE SHC SHE’ 

of minimum, .° LC—(CL+MG)_(LD-~LN)-LD 
: Bvea AZHC = THE 


Cc 


from the principle 


10 Cycloid of Quickest Descent from a Point to a Vertical Line, 


MG LN Ae: , 
/ eas) | 
Or HO WHE? or from ee triangles, 
EL LI EL 
CL./ HO) oD ./ HE Cle aaa 


LED J HE * before, p. 5. 


In Bernoulli’s solution the curve between 4 and Bis 
shewn to be a cycloid; but nothing farther is deter- 
mined ; the result is independent of the relative position 


of the points 4 and B. Hence, the same result, as far 
as it regards the nature of the curve, will obtain for 4 and 
B’, A and B” &c. that is, the curves 4B, 4B’, AB”, &c. 
are all cycloids. A question then naturally arose out of 


tntersects the Vertical Line at Right Angles. 1} 


the original one; Which is the cycloid, down whose 
length a body will descend in the least time ? Or, in other 
words, required the curve, a cycloid it must be, such that 
body shall descend from 4 to the vertical line BB’ B” 
in the least time. This, after the solution of the first 
problem by James Bernoulli, was proposed by him 
to his brother, and solved by the latter, not only 
for the particular case in which BBB” is vertical, 
but for any inclination of that line, and for any 
form of that line; that is, John Bernoulli determined 
the condition to which the cycloid must be subject, when 
the time down it from 4 to a curve DD’ b” shall be 
aminimum. ‘The condition, which hereafter will be 
deduced, is, that the cycloid shall cut the curve at right 
angles *. 


We may now, without either haste or abruptness, 
proceed in the historical and scientific development of 
the new calculus. And, the next step will be to the 
famous Programma of James Bernoulli, which contained 
the problem, whence the title of Isoperimetrical, since 


applied to all problems of a like kind, is derived. 


* This Bernoulli determined by an ingenious, but peculiar method. 
Indeed, there appeared no general method of solving isoperimetrical 
problems, with all their circumstances, till Lagrange in 1760 gave 
his formula of solution, consisting of two parts; the one containing 
integrals, the other a definite expression. See the subsequent parts of 
this Work. 


CHAP al 


[soperimetrical Problems proposed by James Bernoulli — Solved by 
John Bernoulli—His fundamental and specific Equations — Appli- 
cation of them to the Curve of Quickest Descent, and of a given | 
Length—Solution of Isoperimetrical Problems, by Brook ‘Taylor— 


Imperfections of his and the Bernoullis’s Methods. eS 


In 1697, James Bernoulli edited a Programma, which 
proposed this problem * : 


Of all Isoperimetrical curves described on the common 


base BN, to find BEN such, that another curve BZN 


) 
j 
| 
i 
| 


—_— 


shal! contain the greatest space, the ordinate of which 
PZ, is in any multiplicate or submultiplicate ratio of — 


the ordinate PF, or of the arc BF. 


SS ee 


* < Queritur ex omnibus isoperimetris, super communi basi BN 
constitutis, illa BEN, que non ipsa quidem maximum comprehendat 
spatium, sed faciat, ut aliud curva B2N comprehensum sit maximum, 
cujus applicata PZ ponitur esse in ratione quavis multiplicata, vel — 
submultiplicata, recte PF, vel arcus BF, hoc est, que sit quotacunque q 
proportionalis ad datam Ag rectam PF, curvamve BF.” Acta Erud, | 
1697. Mai. p. 214. | 


. 
é 
7 
‘ 


First Solutions of Isoperim. Prob. by John Bernoulli, erroneous. 13 


In the terms of modern analysis, the ordinate PZ 
is said to be a function of PF, or a function of the arc 


BF. 


_ Itis the last case of the problem, when PZ is a func- 
tion of the are BF, that baffled the sagacity and skill of 
John Bernoulli, and caused between him and his brother, 
a long and acrimonious controversy. All the solutions 

_of the former were erroneous *, till the publication of his 
brother’s. And, even when John Bernoulli had amended 
his solution, he would not frankly and plainly acknow- 
ledge his error, but slurred it over, with a faint and 
half-confession of having previously been guilty of some 
slight inadvertencies. ‘The last solution, however, of 
John Bernoulli, published in the Academy of Sciences for 
the year 1718, is, considering what was then the state of 
analytical science, very admirable, and merits the eu- 
logium which he himself has conferred on it, that of being 
equally exempt from the tediousness of his brother’s, 
and the obscurity of Taylor’s calculations}. It is, how- 


* Acad. des Sciences, 1706, p. 235. or Opera, tom. I. p. 424. In 
this memoir he solved the first case of his brother’s problem, making 
two elements only of the curve to vary, but making besides the 

elements of the abscissa to vary. On these grounds the solution of 
the first case is right, but that of the second erroneous. 


¢ ‘ Une voye courte, claire et facile, suivant laquelle un geometre 
d’habileté et d’esprit mediocres puisse arriver jusqu’ avoir, non sur la 
foi d’autrui, mais de ses propres yeux, ces veritez abstruses, sans 
s’engager dans la longueur du calcul de mon frere, ni dans lobscurité 
de celui de M. Taylor.” Opera, tom. II. p. 238, also Mem. de I’ Acad, 
de Paris, 1718. p. 100. 


14, John Bernoulli’s last Solution. 


ever, obviously borrowed from his brother’s, which was 
entitled ‘‘ Analysis magni Problematis Isoperimetrici,” 
and published in the Acta Eruditorum Lips. for May, 
1701, p. 213. Previously to the Analysis, a solution, 
as it was called, but in which results alone were stated, 
had been published in the above mentioned Acts, for 
June, 1700, p. 261. 


As this treatise is not intended to be strictly his- 
torical, the problem of the isoperimetricals will not be 
traced through all its variety of change and emendation. 
To history we shall adhere no farther, than is sufficient 
to preserve an unbroken series of methods gradually 
becoming more exact and extensive; the series begin- 
ning with the first rude, though perfectly just, method 
of James Bernoulli, and ending with Lagrange’s ex- 
quisite and refined Calculus of Variations. For this 
reason we proceed to notice, and shall give the substance 


of the last solution of John Bernoulli, published in the - 


Memoirs of the Academy of Sciences for 1718. The 
separation between that, and the first solution of the 
brachystochrone, which has been already given, being 


neither too wide, nor too abrupt, for our present | 


purpose. 


Bae is acurve, and an element of it abce is com- — 


posed of three portions ab, bc,ce; another element 
aghe, indefinitely near the former, is composed of three 
portions, ag, gi, 1e, and since the solution is to extend 


‘ 


| 


to isoperimetrical curves, we have, 


Fundamental Equation.— Elements of the Abscissa constant. 15 


| ab+be+ ce=ag+ git ie, 
and consequently, (ag—ab) + (gi—bc) + (te—ce) =0 


or, gm—bn—oc+ih=0 [1] 


b 
bmg, abf, gmat . bg 
ke 


By similar triangles, 


any + ae 
ioc, bck, co= F-. Ct. 


ich, "tbey ts mile 
e 


Hence, substituting in the equation [1] 


FO MERG ke bey 
- Ta) bg be ce oh ae 


and this Bernoulli calls his fundamental equation, on 
account of the uniformity * subsisting in the coefficients 


* P. 104. Mem. Acad. Paris, 1718. Bernoulli, after noting this 
uniformity and shewing what it consists in, adds, ‘‘ On verra dans la 
- suite que cette uniformité contribue merveilleusement a reconnoitre 
comme d’un seul coup d’cei] et non en consequence d’aucune analyse, 
’ les equations qui conviennent a chacun des problémes que nous allons 
resoudre,” 


16 Uniformity of Coefficients in the Fundamental Equations. 


of bg and ci; which uniformity, if we run a little before 

our matter, and borrow the symbols of the differential 
- calculus, may be easily made manifest. Thus, fb, ke, 
le, being the differentials of the ordinates Na, Pb, Re 
[y, y’, y' | are to be noted, dy, dy’, dy”; and if dz, d2’, 
dz’ denote the differentials of the three arcs Ba, Bb, Be, 


we have, 
(12-1) te -) c 


dz 


In the foregoing demonstration, the elements of the 
abscissa are supposed to be constant, or the points J, c, 
by whose motion the curve is supposed to be changed, 
move along the lines fg, pc; but, instead of the elements 
of the abscissa, those of the curve may be supposed 
invariable, and this case Bernoulli includes. in ,a second 
fundamental equation, like the former, and demonstrated 
by the subjoined process. | 


Since the elements ab, bc, ec, are supposed to be 


constant, from a, e, as centers describe circular arcs 


Z 


Second Fundamental Equation.—Elements of the Curve constant. 17 


bg, ct, in which, the points b, c, when we pass from 
curve to curve, are supposed to move: now, 
be? ipa 

» bk + ke = (bk +gn+ot)*+(ke—co—bn)? 
or, (bk +gn+oi)’— bk’? =ke? - (ke—co- bn)? 

or, by virtue of the formula, 
| ey! = (2@+y) (ey) 
((2bk+gn+or) (gn+ot)=[2ke—(co+bn)]| (co+bn) 

or, 2bk (gn-+ 02) =2kc (co + bn) neglecting (gn +02)’, 
(co+bn)*; hence, gn+ ot=(co + bn) = a? hd 
afb,bng, gn =f. bn 
cle, cot, ol = ss CO. 
cl 


‘Hence, substituting in equation [2], and transposing 


C- 5) bn- Fy 5) 00 = 0, 


which Bernoulli calls also a fundamental equation, and 


now, by similar triangles, 


in which is to be noted an uniformity similar to that 
pointed out in the former fundamental equation. 


If we express the equation symbolically, we shall 


Ethane dy dy’\ 
Se v) bn — (2g Ie Yi. co= bs 
These equations appeared in Bernoulli's last solution 


of the Isoperimetrical Problems. And to be satisfied of 
the skill and previous labour expended in their con- 


have, 


struction, we need only remark, that they have no 
i D 


- — 


18 Application of Bernoulli’s Method 


dependance on any property of maximum, or minimum, - 
and seem therefore to have no relation to the problems, 
for which they were specially designed. ‘The relation, 
however, which they are made to have, cannot be better 


understood than from the following instances. The 
first of which, is the Brachystochrone, not the common 


case*, but that, in which, to the minimum property, 


the Isoperimetrical is added. In other words, when, out 
of all curves of a given length, the curve of quickest 
descent is required. 


In the subjoined figure let Babce be the curve of 


B_N BS R S 


quickest descent, of which a portion abce is formed of 


three elements, ab, bc, ce; take a portion of another 
curve formed of ag, gi, te, indefinitely near the former, 
then, by the principle of the maxima and minima of 
quantities, the time down ab, bc, ce,=time down ag, 
gi, te. But the time varies as the space divided by 
the velocity varies; therefore as the element of the 


~_- 


# See pages 4, and 9. 


to the Brachystochrone amongst Isoperimetrical Curves. 19 


curve divided by the square root of the vertical height 
from which the body has fallen: hence, 
eee Oc aye enor Ue, Nees. eit 
/ Na /Pb SRe / Na’ / Pn VYRo 


but ab, ag, bc, &c. are all equal ;s 


ME Eth Ripalerinn LEX 
1 ] 1 
PP Pa" 7h 
- 1 r 1 ‘ae 1 ‘ie ] 
7 Sf Pb Sf(Pb+bn)* ~ V(Re—co) SW Re’ 


or, ae (Ce SAE AA OI DL Lite a 
/Pb \VPb  opH’ ” SW Re" oR - Re 
[expanding and neglecting the terms 
involving bn’, co*, &c. | 
Hence, ee = sis and this is an equation expressing 
co Rc 
the proportion of bn to co, obtained from the property 
of the proposed minimum ; but the second fundamental 
equation, involves bn, co, the proportion of bn to co, 
there arising from the Lsoperimetrical condition. Hence, 
eliminating bn, co, there results the specific equation * 


(bea a) Pot — Ae 5) Ret = 0, 


af Ok 
fb ke ke le + 
a, Ce- Bh, er Sik 7) Re 


* “ Specifique ; parceque d’elle yesulte equation differentielle, qui 
determine finalement l’espece de la courbe cherchée.” Acad. des 
Sczences, 1718, p. 108. 


20 Equation to the Brachystochrone. 


Now if we observe this equation, it will appear per- 
fectly uniform in its two members, since the right, hand 
member differs from the left, only inasmuch as the 
points e, c, are used instead of the c, b. Hence, either 
member is always the same, throughout the curve 
required: hence, symbolically, 


d 
ee J Y ot yt = a constant quantity ; 


or, -d () yi = aids 


[dz is constant and multiplied into Wa, in order to | 
make the equation uniform. | 
Hi ts’ dx. bie aey “lt iho dz tra 
du yt 
but dz*=dx*+dy*; and since dz is invariable, o= 
dx.dx + dy dy; hence, substituting in [a] | 
a dx 2 
=k hg) 0 pj hod 
x - 
y 
» Oe 


Or, Ty" “ioe ord (), a2 22n7 a. ae 


and integrating, 5 ——) = eee: +- . ap being the correc™*, | 


dz dx” $y? a b 
Hence, dagen hea ta d/ta! nS 


| d=, 
d, di ORE 0 ala Bm Se. 
Bee Jf (4ac + b% y—CyHAbev ay)” 


* See Emerson’s Fluxions, p. 189. Third Edition. 


ait 


In Isoperim'. Problems, the Variation of three Elements requisite. 21 


If in the correction f, b=0, dz = 2dx Vv: = 

24/ ae and the curve is the common 

| V (4a-y)’ 
eycloid; which now is the curve of the quickest 
descent, not amongst all curves of the same length 
between B and C, but amongst all curves whatever that 
can be drawn .between these two points. This is, 
however, only a particular case, for if b be real} the 
curve is not a cycloid, nor is it any known curve, but 
merely that which the differential equation determines 
it to be. 


If we now examine the preceding solution, we shall 
perceive the office of the fundamental equations in 
solving problems of maxima and minima. Analytically 
speaking, they half resolve such problems ; and furnish 
one equation involving two arbitrary quantities, [bn, co] 
whilst the maximum or minimum property furnishes 
another. ‘That which results from elimination is the 
equation of solution. 


| We may perceive also, in the preceding solution, 

the necessity of making three elements of the curve. to 
_ vary. Two elements are sufficient when merely the 

brachystochrone without any restriction is required. The 

minimum property furnishes one equation, containing 
an arbitrary quantity [LG], which is also the equation 
of solution. If a second property, isoperimetrical, or 

other, be introduced, it may be incompatible with the 
- eurve determined by the equation resulting from the 
minimum property. 


22 Second Instance of the Application of Bernoulli’s Method 


For instance, CGD may be determined such, that it 


oO 


is the curve of quickest descent, and then taking CLD 
indefinitely near, the time down CG-+the time down 
GD=the time down CL + the time down LD; but 
CG+ GD cannot equal CL+LD, for the former sum 
is less than that of any two lines, drawn from the 
extremities C, D, externally to CG, GD. 


This does not shew, beyond controversy, the neces- 
sity of the variation of three elements. For, the points 
L, G, need not lie in the same line ELG, since the 
elements of the abscissa as well as those of the curve 
may be made to vary: in which case, two equations, 
derived from two pfoperties may be obtained; and this — 
method, which was used by John Bernoulli *, in 
certain cases, but not generally, leads to right results. 


If problems involving merely one property, the 


* The method is right in the first case of the problem proposed by 
James Bernoulli, but not, in the second. See p. 13. 


: 


to the first Case of Problem proposed in p. 12. 93 


maximum, require the variation of two, and those in- 
volving two properties, the variation of three elements ; 
problems involving three properties, would, it should 
seem, require the variation of four elements: and _ this 
is the case. Each property furnishes an equation con- 
taining three arbitrary quantities (such as bn, co), and 
the solution depends on the equation resulting from 
their elimination. To such problems, however, the 
Bernoullis did not extend their researches. 


We shall give a_ second instance of Bernoulli's 
method in solving the first Isoperimetrical problem that 
was proposed by his brother; and proceed as in the 
former case; that is, use two fundamental equations 
involving bg, ic, one derived from the Isoperimetrical 
condition, the other from the common property belong- 
ing to quantities, at their state of maximum. Now the 


problem requires the curve to be determined, such, 


that ify be the ordinate, and Y= f(y)*, {¥ dx shall 


be a maximum. 


Hence, since the elements of the abscissa are sup- 
posed constant, [see fig. p. 15. ] 
f (Pb) +f (Re) = f(Ps) + f (Rd, 
by the property of maximum, 
or, f (Pb) — f (Pg) =f (Rt) —f (Re) 
or, f (Pb) — f (Pb + bg) =f (Re- ci) — (Re) 


* f(y) is the method of noting the function of a quantity y. The 
name and symbol were used by John Bernoulli in the Memoirs of 
the Academy for 1706, and 1718. See the latter Vol. p. 106. 


D4, Equation of the Curve. 
from which, expanding f(Pb + bg), f (Re—ci)*, and 


neglecting the terms involving the squares, cubes, &c. 
of bg, ci, we have, 
D.£(Pb) b= DofiRe em 
[p.f (Pb) being the coefficient of the second term, 
in the SERUPR of f(Pb+bg)t | 


If we now substitute for bg, ci, in the fundamental 


sagt ee i 15, there results, 


GA 1 me K Ces EE) 1 
ab ay) D p.f(Pby be sehe’ cones -Ciie) 
In which equation, (specific Bernoulli calls it) the 
law of uniformity being observable in the composition 
of its members, it follows, _ that, 


] ‘ 
(A sii) ot sit is every where the same, 


ean. Beli Da 


pn aaa 100 (G2) hana 


ei : 
: euie from the invariability of the elements of the 


abscissa, constant ; 


dx dy — d’y. dz _ vd.) .dx 


hence, 
dz a - 


* Pb=y, and bg=dy, then, by Taylor’s theorem, [See Princ. 
Anal. Calc. p. 44.) 
f Y+dy)= =fy +t LY dy + n8 > dy + &e. 
or, if we make Y = fy, 2 “a 


dV “1h A 


+ Princ. Anal. Calc. Pref. p. xxix; and p. 43. 


The Circle, amongst Isop', Curves, contains the greatest Area. 95 


but dz’=da*+dy’; ..dz.@z =dy.d'y; since da 1s 
constant, consequently, by substitution, 


d’z (dy —dz)  v.¥. dy.dx 


d z* “a a : 
dz Dae aoe ; 
> dz’ he ST and integrating 
bane Es 


ager ATL EG ey Ae dy=S 7 dite 


fay, and © is the correction; hence, 
: a 


GaP ru), we (Fc)* Ec) 
seedy agus a frien 
din (¥+c) dy 
VJ (a — [Fre}") 
ie f(y) = 1, then Da? = ue ae oy eo. 2. atid. 


dy dy 
(ye) .dy 
fieS Fe [ye 
or c= C— V (a — [y+ c]*) 
an equation to a circle. Hence, of all curves, of a given 
length, that can be drawn between two points, the circle 
is the curve that contains the greatest area. 


If we apply Bernoulli’s fundamental equations to the 
Bi Nes Vp. 8 


second case of the Isoperimetrical problem, {see p. 13.] 
E 


26 Solution of the second Case of James Bernoulli’s Problem. 


that is, when the ordinates are not functions of the 
ordinates Pb, Re, &c., but of the ares Bab, Babc, &c., © 
the process will be somewhat different from the pre- 
ceding. For then we have, instead of 

f (Pb) +f (RO = f (Pg) +f(Ri) 

f (Bab) +f (Babc) =f( Bag) +f (Bag?) 
or, f (Bab)—f(Bab+gm) =f (Babc—co) —f(Babc) © 
and, as before, p.f (Bab). gm=v.f (Babc) .co 


But the fundamental equation imvolves bg, cz; therefore, 
gm, co, must be expressed in terms of them. Now, 


by similar triangles, 


om b CO le cz; hence 
ee We Bb &> Pir aaa: 


pf (Bab) :-* bg =v.f(Babe).~.cis , 


and substituting in a fundamental ues (pets. ae 
Apo ee * 
- bo) TURD = <). ‘le. oF Bab 
Now, Bernoulli remarks, that in this equation, as it stands, 
there is not that uniformity, which enables us to pass from 
one element ab to another bc affected in the same manner, 
and soon. ‘There would have’been the requisite unifor- 


mity, if the factor in the right hand side of the equation, 
bc 
instead ofS es had been ke’ We shall cause therefore 


the requisite uniformity, if we multiply each side by 7, 


and there results the specific * equation, ‘ 


* See Note,’ p. 19. 


Liquation to the Curve. Qo" 
ge “ ie) ab be 1 kc ley be 


"fb ke’ pf (Bab) ~ be” ce? ‘ke’ 


+ SL constant: quantity: | ince 5 
Te'D fF (Babe) *oonstant quantity; hence, airlocpees 
| ae and calling p.f (Bab) = S, we have, 


7 d dx 
-4.( a!) a ays ia? 


ny Db yciayagiis:d a 


.dx; but, dy dy =d'z.dz; 


dz? a dz 
, ay.dx* S§ dy dy S.dz 
ae gr cies ve ee AL, ee A 3 


dx 
Hence, Tags. dz+c)= cae yy: gl JS.dz, 


or, a.dx=dy (Z +c), the ae to the curve. 


This, as we have before remarked, is, in the history 
of isoperimetrical Problems and their connected 
Calculus, the most important problem; since, as long as 
John Bernoulli made two elements of the curve only to 

vary, he constantly, by his erroneous solutions, afforded 
matter of triumph to his brother James. The above 

g olution i is John Bernoulli’s last solution, and an exact 
one ; but, it was not published till 1718, in the 
PTemoirs of the Academy of Paris for that year, six- 
teen years after the appearance of James Bernoulli’s *. 


* This solution appeared in the Acta Erud. Lips. Mai, 1701, 
p- 213. John Bernoulli’s solution (which was faulty) was commu- 
nicated to Leibnitz in 1698, to the Academy of Sciences in 1701, and 
published in their Memoirs for 1706. 


28 John Bernoulii’s-Method distinguished by its Specific Equations. 


On this solution, that of John Bernoulli’s is essen- 
tially founded; and indeed on the very principle, the 
neglect of which had vitiated all his former solutions; con- 
taining also, under the form of fundamental equations*, 
James Bernoulli’s proportions+. The latter are, with 
the slightest trouble, transmutable into the former. Yet 
John Bernoulli, with a total disregard of justice, wishes 
to confer superiority on his method, by the reverse 
operation ; that is, by transforming the former into the 
latter. This, however, is not the sole nor the least trait 
of his unfairness. t 


John Bernoulli's solution possesses greater elegance 
and compactness than his brother’s. Yet these are 
qualities which we may attribute, almost with equal 
justice, as well to time as to genius. It possesses, 
however, a characteristic excellence in the uniformity of 
its specific equations. A considerable advancement was 
thereby made in the calculus of Isoperimetrical problems ; 
and it is on a like principle of uniformity, that cer- 


* Mem. Acad. 1718, p. 103, &c. 


+ deta Erud. Lips. 1701, p.213, or James Bernoulli’s Works, - 
tom. II. p. 899, or John Bernoulli’s Works, tom. II. p. 220, &e. 


{ The palpable error of his former solution he wishes to reduce to a 
mere fault of inadvertence. ‘‘ Pour reparer cette faute d’inadvertence,” 
he says, ‘‘Je vas donner ici une nouvelle maniére de resoudre, &c.” 
Besides, he is ever seeking occasion of aspersing his brother’s method 
[see Mem. Acad. 1718, pp. 102, 103, 131.] and this he does sixteen 
years after James Bernoulli’s death: when, that event, the lapse of 
time, the recollection of his brother’s kindness, a zeal for a broiher’s 
fame, ought to have assuaged and laid to sleep all angry passions. 


Brook Taylor’s Solution of Isoperimetrical Problems. 29 


tain methods of solution, of more recent date, are 
founded*. ~ 


Some of the latter assertions the Student must be 
content to take on trust: for their proof would lead into 


_ too wide a digression. Yet, his curiosity will be amply 


_ rewarded, if he will search for, what may be found, the 


. 


% 


“ 


identity of the methods employed by James Bernoulli 
in his “Solutio magni Problematis Isoperimetrici,” 
and by his brother in his last essay, inserted in the 
Memoirs of the Academy of Sciences. 


With the above-mentioned methods, the researches of 
the Bernoullis on these subjects terminated. ‘Towards 
the period of their-close, in 1715, Brook Taylor, in his 
* Methodus Incrementorum,” solved the problem of the 
Isoperimetricals, on principles not different from those of 
the Bernoullis, but with some alteration of symbolical 
notation. ‘The most material alteration, or rather im- 


provement, consisted in representing the fluxion of V 


7 


* In Lyons’s Fluxions (1758), at page 99, the author, in solving the 


; Brachystochrone, arrives at this equation, 


=. y he aw : 
in which 7, w, are the fluxions of two contiguous ordinates, and V, v, 
the velocities in the arcs. Hence, since there is in this equation a law 


of uniformity, precisely of the same nature as that which Bernoulli 


pointed out ; the author infers, that +7. (¢ the fluxion of the arc) isa 
given quantity. 
The same principle of uniformity serves also, in part, as the basis of 


Mr. Vince’s solutions, Fluxions, p. 195, andof some of Emerson’s, 
pp. 182, 183, &c. Third edition, 


30 Imperfections of preceding Methods. 


when /V% is the analytical expression of the maximum 
property ; thus, 

V = Mi + Ny + Ls 
which mode of expression, Euler, as we shall hereafter 


see, skilfully availed himself of. 


The methods of the Bernoullis and of Taylor, were 
held, at the time of their invention, to be most complete 
and exact. Several imperfections, however, belong to 
them. They do not apply to problems involving three 
or more properties; nor do they extend to cases in- 
volving differentials of a higher order than the first: 
for instance, they will not solve the problem, in which 
a curve is required, that, with its radius of cur- 
vature and evolute shall contain the least area. 
Secondly, they do not extend to cases, in which 
the analytical expression contains, besides «2, y, and 
their differentials, integral expressions; for instance, 
they will not solve the second case proposed in James 
Bernoulli's Programma [see p. 12.] if the Isoperi- 
metrical condition be excluded; for then the are s, an 


integral, since it = J dx / (1 Ne, , 1s not given. 


Thirdly, they do not extend to cases, in which the dif- 
ferential function expressing the maximum should 
depend ona quantity, not given except under the form 
of a differential equation, and that not integrable; for 
instance, they will not solve the case of the curve of the 
quickest descent, in a resisting medium, the descending 
body being solicited by any forces whatever. 


Obscurity of Taylor’s Method. aI 


There are no other peculiarities in Taylor’s method, 
than those we have mentioned, that demand our atten- 
tion. It has no recommendation from its neatness and 
perspicuity, but is justly censured by John Bernoulli 


for its obscure conciseness *. 


We must now direct our attention to a period of 
greater interest, during which, the great Euler, who 
left no part of science untouched or unadorned, directed 
his attention to the calculus of Isoperimetrical problems. 


* <M. Taylor, homme d’esprit, et Geometre tres habile, qui a 
heureusement pénétré jusque dans ce que nous avons de plus profond, 
comme il pardit par son livre de Methodo Incrementorum: sentant 
bien la longueur embarassante de l’analyse de mon frére, et voulant 
la rendre plus courte et un peu claire, a repandu lui-méme tant 
d’obscurité sur cette matiere (aussi bien sur d’autres ou il a voulu etre 
court) qu’il semble y prendre plaisir, et que je doute qu’il y ait 

quelqu’un, quelque pénétrant qu’il soit, qui Pentende partout, quand 
_méme la matiere lui seroit deja connue d’ailleurs,” &c. Mem. Acad. 
1718, p. 103. 


CHA PIT 


Euler’s first Memoir on _ Isoperimetrical Problems — Table of 
Formule — Solution of Problems by it— Methods of Thomas 
Simpson, Emerson, and Mac Laurin. 


An interval of fifteen years had elapsed, before the 
subject of Isoperimetrical problems was resumed by 
Euler. He took it up where the Bernoullis had left it, 
and conducted his first imvestigations on their plan. 
These are not easily intelligible without a previous— 
knowledge of the researches of his great predecessors. 
The Student, who, for initiation into the peculiar calculus 
of Isoperimetrical problems, should resort to Euler’s 
first memoir, would, from the novelty of its terms, © 
principles, and methods, find it perplexed and abrupt. ) 
But, he will fancy himself gliding along the same route, | 
if after Bernoulli, he takes Euler as the guide of his— 


enquiries*. 


Euler’s first memoir was inserted in vol. VI. of 
the Ancient Commentaries of Petersburg for ee 


He there distributes his problems into classes. In_ 


* It is thus with other subjects. Investigation is easy when it is” 
made with the proper series of steps; but difficult, when by our own 
sagacity and labour, we must supply either steps that are wanting, — 
or approximate those which are too widely separate. | 


Euler’s Principle of Solution, the same as Bernoulli’s. 33 


the first, are problems like that of the brachystochrone 
and the curve of least resistance, with the property of 
the minimum, but without the Isoperimetrical condition 
or any other. ‘These are to be solved from the principle 
of the property of a maximum, belonging to the elements 
of the curve, as well as to the curve itself; and from 
the principle of the equality between two proximate 
states of a quantity, when near its minimum or 
maximum *; and they require for their solution the 
variation of two elements only of the curve. ‘Thus, in 
the subjoined Figure which is constructed similarly to 
Euler's +, let CLND be the element that possesses the 


O H 


property of a proposed maximum or minimum: take 
_CMGD indefinitely near ; then Euler says, if we express 
the property in CMGD and in CLND, the difference 
of the two expressions ought to equal 0. 

[a] Now CL is changed into CO=CM+ MG= 


CL4+ LG, 


*See p.7. + Comm. Acad, Petrop. tom, VI, p, 128. 
F 


34 Instances of Euler’s Method. 
[b] and LD is changed into DG= DL- LN= 
LI 
DL- ID’ LG, | 
Hh, LE LI 
or the variations of CL, LD are Cr: LG, - pr Le: 


Hence, in an instance, in which f2%”".ds, is to be 
a minimum (x = abscissa, ds = differential of the curve,) 
we have, 


OA4’.CL + OB".LD=OA".CG+OB".GD, 
hence, by [a], [6], 


Lhe dl 
OA Gy LG=O0B Tp LG: 

eo oe | 

or OA i a Bie OB ‘LD’ 


and consequently from the uniformity of this equation, 
[p- 15, 20, &c.] 


E d } 
O04. , ot x*.52 =a the equation to the curve. 


ds 


d , 
J =a, the equation to a cycloid, 


ds WV/ x 


If n= —4, 


; $8 Ht dg 
and the expression for the minimum is / gee oe fat 


(¢ the time,) or the time is a minimum down the curve, 


whose equation is a. This problem is that 


dy 
ds Ja 
which was solved in pp. 4, 9. and on the same principles ; 
for if we compare the solutions, the two latter will be 
found to difier from the former, merely in the greater 


compactness and regularity of their processes, and in 


—— 


His Classification of Isoperimetrical Problems. 35 
their furnishing something like a clue to the general 
solution of similar problems *. 


The preceding problem is easily generalised ; for if 
P be a function of x, and fPds is to be a minimum, 
we should have, supposing P to become P’ when z 


oecomes x +dx, 


P.CL+P.LD= P.CG+P..GD 
and by [a], [6], : 


LE oy EBigs QO BE 
P.ay LG = P’.5=5,.LG, or Pia = P ‘TD 


an equation, which, according to Bernoulli, may be 
called a specifie equation. 


Hence, P ey =a, the equation to the curve. 

By these methods, hardly to be reckoned different 
from those of the Bernoullis, Euler solves problems of 
the first class: he then passes on to those of the second, 
which besides the property [B] of the maximum, are 
to have another property [4] +; of this class are those 


* The first solution of James Bernoulli, as it stands [see p. 4,] 
affords scarcely the Jeast glimpse of a general method or formula of 
solution. Of its peculiarity, the Editor of John Bernoulli’s Works is 
aware, since he remarks, ‘‘ Quoniam autem synthesin meram parti- 
cularem, nec ad similes casus facile applicandam, &c.” 


+ The following is Euler’s classification : 
** I. Ex omnibus prorsus curvis eam determinare, que proprietatem 
A maximo vel minimo grady contineat. II. Ex omnibus curvis pro- 
prietate 4 equaliter preditis, eam determinare, que proprietatem B 
maximo 


36 Problems involving two Properties, 


problems solved by James and John Bernoulli, in 
which the property [4], is the Isoperimetrical one. 
These require the variation of three elements ab, bc, ces 


and from such. John Bernoulli procures, in fact, two 
equations, involving bg, ci: one equation* called a 
fundamental equation, from the Isoperimetrical property 
{4]; the other + from the property of the maximum 
[B]: and these equations may be thus represented: __ 
[c]...P.bg —Q.ci=0 from Isop'. condit. [4], 
[d]...R.bg -S.ct=0 from max”. condit, [B] 5 


whence results PS=QR, which, if the law of uni- 


maximo vel minimo gradu contineat. III. Ex omnibus curvis et 
proprietate A et proprietate B zequaliter preditis eam determinare, 
que proprietatem C maximo minimove gradu contineat. IV. Ex 
omnibus curvis proprietatibus 4 et B et C singulis aqualiter preditis, 
eam determinare, que proprictatem DD maximo minimove gradu 
contineat. Simili modo quinta classis curvas quatuor'proprietatibus 
preditas contemplabitur et ita porro sequentes.” Comm. Acad. Petrop. 
tom. VI. p. 125. 
p. 15. _ “fp. 19. 


solved by two similar Equations. 37 


formity prevailed in it, as in the two first instances *, 
would be a specific equation: but, if the necessary 


uniformity should be wanting, then it must be intro- 


g; 
duced, by multiplying each side of the equation by 
proper quantities, as was done in the solution of the 
third problem +. Now Euler’s plan and reasoning 1s 
very similar to this: he deduces the equations [c] 
{d]; and then observes, that the quantities Q and S are 
frequently so compounded, that Q=P+dP, and S=R 
+d; or, if not so compounded, may, by multiplication, 
be reduced to that form ¢. 


Euler, in this part, gives no general proof of the 
preceding assertion; but, if it be admitted, then the 
' equation 

QR = PS, becomes 
R(P+dP) = P(R+dR), and consequently, 
| R.dP = P.dR,.or oe ae and integrating 
P+aR=0, the equation to the curve. 


In point of practical convenience of solution, a great 
step was made by Euler in the preceding process. For 
the computist is directed to attend solely to the deriving, 
from one of the proposed properties, an equation of the 
form, 


P.bg — (P+dP) ci; 


mpp.20,24.~ + p. 26. 

t “In quibus quantitates 2 et S plerumque ita sunt comparate, ut 
sit 2=P4dP, et S=R+dR. Si vero hujusmodi formam non 
habuerint, poterunt semper multiplicando vel dividendo zequationes 
ad talem reduci.” Comm. Acad. Petrop. tom. VI, p. 134. 


38 Resulting Equation of Solution: 


whence FP will be known. He must then derive from 


the other of the properties a similar equation, 
R.bg —- (R+dR).ci 


whence R will be known: and the resulting equation 
of solution will be 


PTaR: 


For instance, suppose one of the properties [B], to 
be such, that fy”.dx shall be a maximum; then, since 
dx is constant, 

Na’ + Pb" + Re” = Na’ + Pg? + Rv 
and «. Pb” — (Pb + bg)” = (fic — ct)" — Re*, 
anu} i Of = Wee. ces 
or PO be = he. cu= 0, 


which, since Rc is the contiguous or consecutive ordi~ — 


nate to Pb, and therefore (if Pb = y)=y+dy, is of 
the proposed form, 
P.bg — (P+dP) ci, 


in which Pay =9%, or to render the equation 


homogeneous, y”—'.dx; similarly, if 7’ had been a func- — 
tion of y and the property f’7'.dx; P would have been — 


eed 


dy dy 


or ES ates Suppose the other property [4] to — 


have been fx”".ds=a, a being constant and ds the — 


element of the curve, then 


BN”.ab+ BP”.bc+ BR”. ce=BN”.ag + BP”. gi+ 


BR” ie, 
and substituting according to the forms [a], [6], p. 33. 
there will result 


: 
: 


a 
, 


A 
; 
, 


Application of it to Instances. 39 


BN _ Bp» = be — Bp™.5— BR") .¢ 
ab 


=0*, which is evidently under the proposed form, 


“aif we make R= BN”. ie B pat, or symbolically = 


a”.dy 


d Aye s Similarly, if instead of /'v” ds, the form had 


been f.X.ds, Xa function of «x, then there would have 
resulted, 


The equation therefore to the curve with these two 


properties, 4 and Bb, 
di ey 


or generally, ~ dx-+a.d ee : 


ae 


is y">'.da 


Since the equation P.bg —- (P+dP) ci, derived 
from the property B, is precisely similar to the equation, 
R.bg — (R + dR) ci, derived from the property A ; it 
is plain, that the resulting equation, that which deter- 
mines the curve, or the relation between y and a, will 
be the same, if B and 4 were changed; that is, if 

P.bg —-(P+ dP) ci 
be derived from A, and 
R.bg — (R+dR) ci 


* This is exactly under the form of John Bernoulli’s specific equas 
tions, for the law of uniformity is manifest in it. 


40 Commutabihty of Properties. 


from B. For instance, suppose the curve to be required 
which, with a given length, should contain the greatest 
area; here the maximum property Bis fydz, and the 
Isoperimetrical property 4, is fW(da’ + dy’) =a. 
vdx 


to a circle: which equation would equally result, if we 


The resulting equation is dy= an equation 


required the curve which; with a given area, should 
contain the greatest arc. 


This important remark of the commutability of the 
properties was originally made by James Bernoulli *. 


Euler, we have seen, reduced problems of the second 
class to a dependance on two similar equations of the 
form 

P.bg-(P+dP) ci=0 
the determination of P depending on the proposed pro- 
perties : for instance, if the property 4 or B were fTidx, 


T=f(y), P would equal —— thy or Fda, If the pro- 


SSS eee 


«Tn guibus omnibus singularis quedam observatur reciprocatio. 
Quemadmodum enim, exempli gratia, inter omnes figuras ejusdem 
Perimetri circulus maximam possidet aream, catenaria maximam 
conversione sui gignit superficiem, solidumque maximum elastica ; 
sic inter omnes vicissim figuras quz aut zqualibus gaudent areis, aut 
equales rotatione gignunt superficies, solidave equalia; circulus, 
catenaria et elastica minimo clauduntur ambitu, quod pariter procedit in 
omnibusaliis.” Acta Erud. Lips. Mai, 1701, p. 213. or Opera, tom. IT, 
p- 919. or, John Bernoulli, tom. IT. p. 234.— Euler makes the same 
remark. Comm. Acad. Petrop, tom. VI. pp, 135, 150. and tom. vill, 
p- 175, and in his Methodus Inveniendi Curvas, &c. See also 
Emerson’s Fluxions, p. 170. 


eo > 
See ee 


Euler’s Table of Formule. 41 
ry Pere 
perty were f T.dy, T=f(x), P would og OF de. 


If the property were (T.ds, T=f (x), P would equal 
d (rn) ; and by observation on the resulting forms for 


P, Euler generalised his conclusions, and arranged them 


in a table, after the manner of the subjoined specimen *: 


Proprietates Valores Litteree P 
proposite. respondentes. 


co de.) Ole Mdyt i. Ba de 
my .ay.. oF] Nda:.: 4 R= NM de 


d 
Gftds... dT — Naz. Bad (P=) 


w. fT .ds...dT=Mdy... P=d. (7) - Mas, 
. &e. 


and of these forms he gave fifteen, by reference to which; 
any problem belonging to the second class might be 
solved. 


s 


For instance, suppose the curve to be required, which, 
amongst all others of the same length, should contain 
the greatest area. Here, 


* Comm. Acad. Petrop. tom. VI. p. 141. 


+ dT=M dy, and & = , or M is the differential coefficient of 'T, 
“4 


ee ae a hy 
making in TJ, y to vary. Similarly, N=7> is the differential 


coefficient making in 7', x to vary. If T should contain both x and y, 
that is, if dT=Mdy+ Ndx, then M and N would become partial 
differential coefficients, See Princ. Anal, Calc. p. 79. 


G 


42 Application of the Table of Formule, 


the maximum property B, = fydz, 
the Isoperimetrical 4, = fds = a. 
By Form 1. =e t, =, P= ae 


By Form 1m. T= 1; P= a(S) o rR= a(S) 


Hence, since the equation is P + aR=0 [see p. 38.] 


dvta.d ($4) = 0, or & pad -— ¢=0%c = cor- 8 


rection) ; and by reduction, 
(v—c).dx 
Again, suppose the curve to be required, which, 

amongst all others of the same length, shall, by a rotation 


round its axis, generate the greatest solid. Here, 


B= fy .dz;... by Form iaT = y, M=2y, P=2yday 


dy = 


an equation to a circle *. 


A= Pus ; ”. by Form 11. P or R =a(% 


Hence, 2y. de + a.d(S 2) =0, 
ds.d*y—dy.d’s 
ds° 
But since dx is constant, and ds*=da’*+dy’, ds.d’s = 

dy.d’y; therefore, substituting, 


y ax? 
ass 


or, 2y.dx+a. = 0; 


oy.de+ at =—1 6 


* See Emerson’s Fluxions, third edition, p. 187; also Simpson’s 
Fluxions, p. 485. 


- 


to the Solution of Problems. 43 
a.d’y.dx 
(dx* +dy*)* 


integrate, and we have 


and, 2y + = 0; multiply by dy, and 


o a.dx 
y? — —————— = ec _[c=correction, | 
(da* + dy’)? 
(y*—c) dy 
v (a — Ly - es 


elastic curve; and which in a particular case, when 


whence, dx = an equation to the 


c=0, becomes 
teak at a 
~ 7-H) 
and the curve in this case is ee the rectangular elastic 


curve * 


As a third example, let the curve be required, which, 
amongst all others of the same length, shall have its 
center of gravity most remote from the axis. Here, 


(calling x the distance from the axis) B= if: z SS fo) BY. 
Form 11. [since s is a given quantity | P=d(«%) 
again d=fds; «by Formm. P, or R=d (fuss Ee -“) 

dy dy 
Hence, a.d(>*) +4(23 2) =0; 


Z (a2) P=, or cds=(a+a).dy 


an equation to the catenary. 


* See Simpson’s Fluxions, p.486, where the solution is not 
general. 


44, Problems of the third Class, | 


This example could not have been solved by Euler’s 
table, if the property had been any other than’ the 


Isoperimetrical : for s, an integral, =fdr/(Q Ne 


and Euler gives, in this memoir, no general method of — 
finding the resulting equation, such as P is in his table, 
when the analytical expression of a property involves 
integrals. See tom. VI, p. 144. 

By means of this table, the practical solution of Iso- 
perimetrical problems, was, as it has been already said, 
very materially expedited. Ina subsequent part of his 
memoir*, Euler increases his table by nine new forms: — 
making the whole number twenty-four. And although | 
this table is now superseded, yet its examination is not | 
without interest, simce we may discover in it the parcels — 
of that general formula, which the author afterwards ex-_ 


hibited. | 
| 
: 
‘ 


Having given rules of solution for all problems 
belonging to the second class, Euler passes on to those 
of the third. In this a curve is required, which, 
amongst all others equally possessing the property 4 and ] 
the property B, contains a property C of maximum or 


minimum. For instance, if the curve be required, 
which, amongst all curves of the same length and the 
same area, is such, that the time down it is a minimum, | 
This class requires the variation of four elements of the 


* Comm. Petrop. tom. VI, p. 146. 


solved by Three similar Equations. 45 


curve; and consequently the methods of the Bernoullis, 
without extension, are insufficient. ‘The course pursued 
by Euler is like his former one in the second class. He 
constructs a diagram similar to the one, p. 36, the sole 
difference. arising from the introduction of a fourth 
element of the curve; and he reduces the analytical 
solution to a dependence on three similar equations 
instead of two. 


The general form of these equations is, 
P.bg-—Q.ci+ R.dd 
dd* being a variation of the ordinate similar to the varia- 
tions bg and cz; and he makes an observation similar to 
the one in p. 37; namely, that it frequently happens in 
simple cases, that Q=P+dP, R=P 4+ 2dP+dP; 
but, if such should not be the form of Q and R, then 
the skill of the analyst must be directed to reduce them 
under that form. | 


When three equations, such as 
P.bg —(P+dP)ci+(P+2dP +a@P).dd=0 
p.bg—(p+dp)ci+ (p+2dp +d p).dd?=0 
r.bg —(r + dr)ci+ (a + 2dr + d’x).dd =0 
are obtained, the resulting equation to the curve will be 
Pia ae bate: 0 » {1 
for, taking the differential, dP +a.dp+b.dr=0 [2] 
| and again, d@’P+aap+b.dr=0 [3] 
And since, [1] x bg=0; ({1]+[2])xct=0, and 
((1]-+2[2] + [3]) xa = 0; 


* dd has no connexion whatever with the separate symbols d, 3 
+ Euler, Comm, Acad. Petrop. tom. VI, p. 149. 


46 Instances of Problems of the third Class. 


P.bg + ap.bg + br.bg =0''9 
—(P + dP).ci — a(p + dp).ci — b(r+dzr).ci = 0 
(P + 2dP + dP) .dd+ a(p + 2dp + d’p)dd + b (4 
2dr+d*r) dt = 0; 

which equations, adding the quantities that are placed | 
vertically, verify themselves; and accordingly shew, | 
that the equation 

P+ap+br=0 
satisfies the three equations. | 


The solution of problems then would be similar to that — 
of those in the second class, and would be immediately 
obtained, if the quantities P, P> 7, were contained in his | 
table of forms *; and in many cases they are. For 
instance, suppose the curve to be required, which 
amongst all other curves, of the same length and same 
area, generates, by a rotation round an axis parallel to 
the ordinate y, the greatest solid. Here, 
the max". property Cis fa°dy; .. by u.p.41. P=2 dz, 
the: «<property, «4.18.9 danza cerby ennp = ala, 


the . property B is fds Osby Tues = “Ge 


and eG the equation to the curve is 
OR i 
aede + a.dv + b.d (52) 0; 


2 . . d q 3 
and integrating, «7 + ax +b = =c .[e = correction ]§ 


(a* + ax — c) dx 


whence dy = J (6 — [ae + anc} ° 


* Comm. Petrop. tom. VI, p. 151. . 


ti 


Objects attained by Euler in his first Memoir. 47 


Again, as a second instance, let us take a third case of 
the brachystochrone, of which we have already had 
two, pp. 4, 18; that is, besides the condition C of quick- 
est descent, let the conditions 4 and B be those of a 
given length, and a given area ; 
here, C = oe . by ur. P=d Givi) 

A= fydz ; ey Teich 


= as “bY II. Or = wt 


I 


and usiears the equation to the curve is 


On ny) +adx +b. a(SL) =0; 


and integrating, er. ot + ax + psu 1 o 


Euler, as we have observed, commenced his researches 
where the Bernoullis had terminated theirs; but, at the 
end of this his first investigation, he was considerably 
removed from the original point of departure. Several 
important objects had been attained by him; the 
solution of problems involving three or more properties ; 
the reduction of such problems to a dependence on two 
or more similar equations; the solution of problems of 
the first class, and of some of higher classes, by a more 
general method*, and by reference to a table of 
formule. 


These methods were held, by their author, to be so 


* Tamen eas methodo paulisper diversa et multo latius patente 
sum persecuturus, &c. Comm. Petrop. tom. VI. p. 127, 


48 Imperfections of Euler’s Method. 


complete, that, on the ground of facility, nothing farther 
was to be desired*. They are, however, not exempt 
from several imperfections. Problems, involving the dif- 
ferentials of x or y of an higher order than the second 


cannot be solved by them: for instance, that which 


: dy 
requires, amongst all other curves, one, in which 3: ae dy 


is a maximum or minimum. Secondly, problems cannot 
be generally solved by them which involve integrals, 
such not being constant: for instance, that in which it 
should be required to find a curve that amongst all other. 
curves, has its center of gravity lowest. Euler solves 
this problem when another condition, that of the 
Isoperimetrical property, 1s added: for then the are s, the 


integral of dx/ ¢ +22) , 1s in all curves supposed to 


be the same --. 


The researches of Maclaurin+, Emerson $, and 
Simpson ||, on this subject, may here be noticed. With 
regard to practical methods of solution they do not extend 
so far as those of Euler, which we have been speaking of ; 
and in point of perspicuity, if we except Maclaurin, the 
other two mathematicians are inferior to the learned 


foreigner. 


* Atque methodo tam facili solutum ut nihil ampliuse desiderari_ 
posse videatur. Comm. Petrop. tom. VIII, p. 159. | 


+ See p. 43. 
} Fluxions, p. 478. § Fluxions, Third edition, p, 170. 
| Fluxtons, vol. II. p. 480. Third edition, 1750. 


Maclaurin’s Formula. ~ 49 


The methods of Maclaurin and Simpson (for Emer- 
son’s is plainly taken from that of the former) extend to 
eases in which more than one property is involved: but 
they are inapplicable to the three cases, and the connected 
problems enumerated in p. 30. 


Maclaurin’s formula of solution is this: If X and Z 
are functions of w, then if Xds~ Zdy be a minimum or 
maximum, Ady =Zds.. This result is included 
amongst Euler’s. [See Comm. Acad. Petrop. tom. VI. 
pp. 141, 142, 143.] For since Xds expresses one pro- 


perty, and dX= oF aa, or since X is a function of 2, we 
have, by Form 11. the quantity corresponding to P 
[see p. 41.] =d (x WY), and for Zdy expressing the 
other property, by Form 11. the quantity, corresponding 


r. 


Z iA 
to P,=——.dx; consequently, the resulting equation is 


dx 
dy, & 
d (x =) —a.dZ and Xdy=aZds, 
the same result in fact as Maclaurin’s. 


Simpson’s method is equally restricted with Mac- 
laurin’s: it rests too on the assumption of the principle, 
that the property of minimum or maximum, true for the 
whole curve, is true also for any portion of it. The want 
of generality, therefore, in this principle, would vitiate 
the method in its application to the excepted cases. 


The methods just described solve not problems of 
greater depth and intricacy than those of the Bernoullis ; 


‘although, it must be remarked, they are invested with 
7 H 


/ 


50 | Simpson’s Method. 


greater analytical neatness and compactness. ‘They are 

not, however, more perspicuous; and, even if they did — 
possess greater extent and clearness, it would not suit 
the purpose of the present Tract, longer to insist on them, 
since they conduct us not towards that formula and 
algorithm, with which the researches on this subject. 
have been closed. - 


The plan of this Work now leads us to theconsideration — 
of Kuler’s second memoir; in which, he very materially © 
improved the calculus of Isoperimetrical problems; 
although he introduces his resumed researches with a 
prefatory remark, that his former methods of solution, 


on the footing of facility, left nothing farther to be 
desired *. 


* Atque methodo tam facili solutum, ut nihil amplius desiderari 
posse videatur. Comm. Petrop. tom. VIII, p. 159. 


CHAP. IV. 


Euler’s second Memoir-—General Formule of Solution—Characters 
of distinction, which Problems admit of—Exceptions to the 


general Formule. 


Tue formation of the table* in Euler’s former 
memoir depended on this principle: If (/Vdz is the ana- 
lytical expression of the maximum property +, certain 
functions of 2x, y, &c. are substituted for ¥, and thence 


* Pp, 4. 


+ Since in the succeeding part of this Tract f/Vdx will be frequently 
| used as the analytical expression of the maximum property, it may 
‘not be improper to illustrate its meaning by one or two examples. 


: ds vis 
In the brachystochrone, the time = Sf aH = @ minimum; therefore 


' dy? 
axrnf/ ( ]+-—— 
ds _ pv (dxetdy') _ de) | 
f V5 =f ve =f- 7Y oe which, compared with 


vate oe) ) L047) pad 


JS Vdx, gives V= re 


Tal 


3 
Inthesolid of Jeast resistance, the resistances f d ee SS x3 


p® 
ee 

If, with a given area, the curve generating the solid of least surface 
is required, then [see p. 37.] fVdr=/(2ry.ds + aydx) = 
S(27yf(1+p?].dx+aydz); «. V=ay+2ryV(1+p"), 

And from these expressions, the value of dV may be found by the 
common rules of the differential or fluxionary calculus. 


therefore, comparing, V = 


52 _ Expression for the Differential of the Maximum. 


é 
a resulting equation (according to Euler the equation — 


P =0), is obtained. In the present memoir, he represents 
the differential of V, or dV, as Taylor had done [p. 30.] 5 
that is, he puts 
dV = Mdx + Ndy + Pdp + &c. + Lds* 
[p= a , and s = arc, | 

and from this more general formula of representation, he 
deduces an equation comprehending almost all his former 
particular equations which he had registered in a_ 
table [p. 41.]. His mode of proceeding is not materially 
different from his former mode. If V contains no~ 


integrals, but is merely a function of a, y, dy, &c., then 


the property expressed by {dx is common to the 
curve, and to its elementary portions: hence, if we 


suppose two portions only of the curve to vary, and if 


Vdx corresponds to CL, and V’.dx to LD, (V+V’) 


* Pin this and the following pages is merely a coefficient, and 
different from the P of the preceding Chapter. - | 


Euler’s Method of estimating it. 53° 


dx isa minimum. Hence, that peculiar differential of 
(V + V’) dx, which arises from changing CL, LD, into 
CG, GD, is equal to nothing: which differential is easily 
obtained from the common differential, since each has this 
in common, that of being the first term of the difference 
of two successive values: now the common differential, 
or, (dV +dV’) dxr= 
(Mdx+Ndy+Pdp).dx+ (M dx+ Ndy'+P'dp’) dz, 
supposing the differential of V’ to consist of three 
terms only, and MM’, N’, P’, to be the values of WM, N, 
P, when V becomes V’’. Now, in order to compute the 
changes in the several terms, we may observe, that since 
AC=y, and since dx is supposed to be constant, the 
terms Mdzx, Ndy, are not affected by the translation of 


the point LtoG; but Pdpi is; forp= riz =CR: but when 
Li is transferred to G, p= ao = = Bees conse- 


quently the variation produced in p, or what dp 
| becomes, when it expresses the peculiar differential 


3" Tee ta 
_ which we are seeking, is CR *. Hence the term Pdp 


SE SES SSR _ Rea at Re RR RRR 9 URSA = SREY S RES DRA EE OE ARR SET NR arm renee, oe, 


| * Those who are possessed of the calculus of variations will here 
_ perceive the very great advantage accruing from the mere in- 
vention of a symbol 2 to denote an operation, bordering on, yet distin- 
guished from, the operation which d denotes. Circumlocution and 
ambiguity were both rescinded by thatinvention. It is wonderful that 
Euler did not hit on it: for the rules belonging to such a symbol, he 
plainly lays down in Comm, Petrop. tom, VIII, p. 163, and afterwards 

more 


54 Formule of Solution. 

must be written P.— , putting »=LG, and dx=CE. | 
Again, y’= BL, and by the translation of L to G,= BG 
=BL+». Hence, dy’ must be written », and NV’. dy’, 


N’v. again p’ = — and by the translation of L to G= © 


LI-~LG_LI-y 
dx edz, - 


Hence, dp’ must be written ee 


and P’.dy’, TE -. Coilecting the several terms, we have 


the whole variation of (V + V’’) dx equal to 


(P. = +. Vy 5). dz, or (N'- ——) v.dx, or 


dP bia a 
(V — o—) ».dx, retaining quantities of the same order : 


lastly, since the variation must equal nothing, we have for 
cases comprehended within the expression dV = Md¢z 
+ Ndy + Pdp, this formula of solution, 
dP 
N- Ss aE 

‘By a process similar, but longeron account of the 

introduction of the quantity ds, Euler deduces from 
dV =Mdz+ Ndy + Pdp + Lds 
this formula, of solution, 
_aP  Léy 


dx zs ds saan 


more fully and distinctly in his tract on Isoperimetricals; and in his 
Jast memoir, Novi Comm. Petrop. tom. X. p. 54, he says, ‘* Illud 
problema Isoperimetricum latissimo sensu acceptum, prout id quidem 
in libro singulari pertractavi; quem qui attente legerit, non dubitabit, © 
quin hujus generis investigationes calculi speciem prorsus singularem 
postulent, a consuetis analyseos regulis non parum diversam,” 


Problems of the second Class. 55 


The two preceding formule have been deduced for 
the first class of problems that involve one property only. 
But Euler’s method, although more tedious, is not essen- 
tially different for problems of the second class. These 


require the variation of three elements of the curve 5 


and accordingly, we must compute the variation in 


) i « 

(74+ V’+V") dx, when the three elements, instead of 
ab, bc, ce, become ag, gi, ie; that is, as before we 
must note the variations in 


M.idz + N.dy + P.dp + &c. 
M’ .dx+ N'.dy'+ P’.dp' + &e. 
M" dx + N”.dy’ +P". dp" + &e. 


arising from the translation of the points b and c, to g 
and 7, and from the introduction of two arbitrary quan- 


tities or variations such as bg, ci, and thence will result an 
| 


equation of the form 4.bg— B.ci=0, which compared 
€ 


with the equation, R.bg — (R+dR).ci=0, which Euler 


56 Formule for Problems of the second Class. 
Be 


had previously obtained *, would give oe 


Rk a4 


whence Ris known. 


» 4 


In the formula dV = Mdx + Ndy + Pdp (forming 
the equation 4.bg —B.ci, according to the precepts i 


given,) 
A=WN .dx — dP 
B es N4,.dx rs dP’ 
consequent ys ee = ee and R= N’dx— dP, 


or = dx Gigs 


This is an equation derived from one property fVda: 
deduce a similar equation from the other fWdz; for 


instance, such as e=dx (> _ =) , and the equation to 


the curve will be 
: d 
R+ae=0, or (w= \ata(eE2) SNAPS 
or N-+ ay — sit Ne = 0% 


& 


and since this form is like that of W— ued , Instead of two 


de 
operations for finding the values of R, R’, from [Vi tee 
a maximum, and f Wdzx = a constant quantity, we may 
substitute one, and deduce the resulting equation fro 


| 
| 
| 


% PR: $7 


Similar to those for Problems of the First Class. 57 


f[Vde+afl War, 
or [(V¥ +a) dz, 
and for this great simplicity introduced into the calculus, 
we are indebted solely to Euler. 


For the sake of stating and illustrating Euler’s 
method, we have taken a simple form to represent dV, 
that is, Mdx+ Ndy+Pdp: but Euler introduces a term 
Lds, which, for problems of the second class, renders the 
calculation longer, and the result more complicated: in 
this case, however, 

dR _ ~@P+U'dx.de+dud(L'e+ N) [= 
R- —dP+dz (L': +N) ds 


a Lide.ds d{-dP +dx(L:+N)] | 
~ —dP+dx(Let+N) | —dP+da(la+N) * 
J Ldxz.dé¢ 


and, R = e~?*e G4 y (_ dP+dx [Le+M)]). 


This memoir of Euler contains almost all the matter 
which is to be found in his subsequent tract on [soperi- 
metricals: but, the matter is ill arranged: considerable 
‘perplexity, and some error, is introduced by the term 


‘Lds: the cases in which it enters, are mixed with those 
in which L=o0. These latter, in the above mentioned 
tract, the author separately considers ; and even here he 
notices, that as from dV =Mdx+Ndy, dVv=Mdz + 
‘Ndy+Pdp: the resulting formule of solution are, 


N=0, and N-&=0, so if we put 
dV= Mas + Nay + Pap + Qdq+ Rdr+&e., 
da dp d 
| Y q= ~, a i » &e. | 


then the resulting Cae will ¢ 
I 


58 General Formula. 
dP #@Q &R 
NTR aes Lae 


which, by the principle stated in page 52, will belong | 


to problems of all classes involving definite expressions, 
This last formula, remarkable for the simplicity of its 
law, supersedes Euler’s table of the values of P*, given in 
his former memoir. In fact, all problems, in which 


integral expressions do not enter, are solved by it; 


and although Euler himself, and afterwards Lagrange, — 


very materially simplified, and expedited its proof, yet, 


as a formula of solution, it still remains as a final | 


result of all researches on this subject. 


Problems are classed according to the number of 
properties which they involve +; and the number deter- 
mines that of the ordinates that must be made to vary. 
But, problems might be distinguished alsofrom each other, 
by the order of differentials which they involve; and 
then the order would determine the number of terms to 


be used in the formula dV =Mdx+Ndy+Pdp + &c. 


thus, ifthe curve were required, in which f (ax —y*)ydz 


is a maximum, we have V = (axr—-y’)y, dV =~ 


ay .dx+(ax—3y’) dy: consequently, two terms are suf- 
ficient, namely, Mdx, Ndy, and the formula of solution is 


Vi =O} or Wes S year. 


This problem is, according to Euler’s classification, of — 


the first class, and might be said also to be of the first — 


order. 


* See p. 41. + See p. 35, 


Distribution of Problems into Classes and Orders. 59 


If the brachystochrone, with no limitation, were 
oni since ee time 


| al Sf nw) Big vite 


by comparing it ag ben we have V= 
_M(i+p’) | 
2 y 


vel’, y 
V(i+p’) y 
Vy 


and thence dV = dp; so 


alii Ug 7 +p’) 
that, the formula Mdx+ Ndy+Pdp, or three terms 
are sufficient, one of which Mdx=o. 


This problem of the first class, might be said to be 
of the second order. 


Again, the solid of least resistance is of the same 
class and same order with the preceding problem ; for 


3 
since ee a minimum = f Vdx, by comparison, 
we have, = ae and =P. dy ‘ar -y dp. 


But, ifa curve were required, in which fo re should 


be a minimum, then since Vd2=St = ; iid 
V=——"~ , anddV= - 5 ide .dp + 
Tine P) (1 +p)? oyAtEs a) +p) ** 


so that this problem would require four terms, or this 
formula 


dV =Mdx+ Ndy+Pdp+Qdq; 
(in which Mand N=0), and therefore might be said to 


*® See p. 51. 


60 The Class determines the Number of Ordinates that must vary. 


be of the third order, and, according to Euler’s classifi- 
cation, would be of the first class, if it depended on the 
minimum property alone. 

The class of problems then determines the number 
of ordinates that must be made to vary; but, on what 
might be called their order, would depend the 
number of ordinates to be taken account of, in the 
computation of the variations. For, as more terms of 
the series Mdx+Ndy+&c. enter, more ordinates are 
required to estimate the variations of the terms: for 
instance, let dV= Mdx+-Ndy+Pdp: then if the pro- 
blem is of the first class, the variation of one ordinate is 
sufficient. Let * BL =y’ be the ordinate that varies, and let 
the two adjacent ordinates be y, y”, then it is plain, from 


the subjoined table, that three ordinates y, y’, y” are suffi- 


cient for the computation of the variations in 


Mdx+Ndy+Pdp, andin Mdx+ Nady +P’ .dp’, 


dy EL pte b aa y 
y = Aer kip = At a Variation in p = FX 
dy’ LI wn v 


variation in py =— 


Pe BW Re de 


cede 
y= KD | P= a 
vy 


The introduction of other ordinates, such as y”, y™, 
&e. or yy, &c. on either side, is plainly unnecessary. 


If, however, dg=—% oP Su enters into the form, or if 


x das 
dV = Mdx + Ndy + Pdp + Qdq, 


e See Fig. pay: 


dx’ 


The Order, the Number that must be brought into Computation. 61 


then three ordinates are not sufficient: an additional 
one y. must be taken account of: for the element 
Vdx of the area analogous to Vdx, and immediately 
preceding it, is Mdx + Ndy, + Pdp + Q.dq; 
now, the last term Q dq, of this formula, is af- 
fected by the increment » of the ordinate y’; for, 


, SG heck aoa ‘aimed i. Oa : 
ee 9 RS tS oe IS varied, if p is 


varied; and p is varied, if 7 is. Hence we must intro- 
duce into the computation an additional ordinate y, ; but 
that is sufficient; for none of the terms representing 
V" dx, V".dx, &c. will be affected by the variation of 
the ordinate y’. 


What has been said relates to those cases in which ¥ 
involves no indefinite expressions; for if a term, such as 
Lds be introduced, a fourth ordinate must be taken 
account of. Ifthe problem is of the second class, then 
two ordinates must be made to vary, and the number of 
ordinates that must be taken account of is easily deter- 
mined from the preceding reasonings. We shall have 
occasion, however, to notice this case, when we speak of 


Euler’s treatise on Isoperimetrical problems. 


In the conclusion of his memoir, Euler notices, that 
his methods fail, or require peculiar artifices, if the 
quantity V in fVdz, contains the arc s, or other inte- 
gral quantities. For, in that case, the principle of the 
whole curve possessing a certain property of maximum 
or minimum, if a portion of it does, is not true. For 
if V contains an integral expression, then the curve 
OCD may contain fVdx a maximum or minimum, 


62 Exceptions to Euler’s Method, caused by Integral Expressions. 


although no separate portion of it has that property. If, 
indeed, 


dV = Mdx+ Ndy + Pdp + &c. 


Me d 
then, since the quantities 2, y, =a or p depend on the 


point C, the part CLD may be changed at pleasure, 
without any change being produced in OC; but if V 
contains an integral, then if CLD is changed, OC will 
take another value from fV dx. In these cases then, 
the preceding methods of Euler, deduced from assuming 
a portion of the curve possessing the requisite property, — 
fail: so likewise do the methods of Bernoulli and 
Taylor, which are founded on the same principle *. : 


There are cases, however, Euler observes, in which 


- 


* Bernoulli’s has been already mentioned; and Taylor, in his — 
Methodus Incrementorum, p. 67. prefixes to his solution, a similar one. 
Its truth depends, as it has been said, on the two portions adjacent to 
CLD not varying whilst CLD varies, which is to be granted, when | 
SV dx contains no integral. % 


Objects attained by Euler. 63 


the ordinary solutions may be applied; although the 
quantity {dx contains an integral: and these happen 
when the integral is by the conditions common, or the 
same to all curves, amongst which a required curve is to 
be found. Included in these cases, are the Isoperimetrical 
problems of the Bernoullis; for there s, the integral of 


dx 7 ( 1+ =) is given: had it not been given, or if, 


instead of the Isoperimetrical property, another had been 
substituted, the solution of the second case proposed in 
James Bernoulli's programma, would have been faulty. 


Very important objects were obtained by Euler in 
this memoir. The solution of problems involving differ- 
entials of any order; the invention of a formula includ- 
ing his former formule, which, to the number of twenty- 
four, he had inserted in a table ; the partial solution of 
problems involving integral expressions; the establish- 


ment of his theorems and formule by easier pro- 
‘cesses *, 


An author is usually, more than justly, fond of his last 
inventions: and Euler, by this memoir, thought he had 
nearly perfected the method of solving Isoperimetrical 


* “Non solum faciliorem quam ante detexi viam ad _ solutiones 
hujusmodi problematum perveniendi, sed etiam omnes 24 formulas, 
quas ante tractaveram, in unicam sum complexus, &c.” Comm. 
Petrop. tom. VIII. p. 159. 


* 


64 Imperfections of his Methods. 


problems *: yet his methods were not without their 
imperfections. They afforded no general solutions o 
problems involving integral expressions; and, erroneous 
solutions, when the differential function depended on 
a quantity given solely by a differential equation not 
generally integrable: and the cause of these imperfec-_ 
tions was the assumption of the principle, that the 
whole curve will be endowed with the property of max- 
imum or minimum, if any portion whatever of it possess 
the same [see p. 52, 62. ]. 


We are now led, both according to the order of his-_ 
torical and scientific succession, to the consideration of — 


Euler’s tract on Isoperimetrical problems. 


* « Triplici ratione illam priorem methodum ad majorem perfec. 
tionis gradum sum evecturus.” Comm. Petrop. tom. VIII. p. 175. 


! 


CHAP, cV. 


Euler’s Tract, entitled “ Methodus inveniendi LineasCurvas Proprietate 
Maximi Minimive gaudentes”’ — Distribution of Cases into absolute 
and relative Maxima and Minima—Rules for finding the Increment 
of Quantities dependent on their varied State — Formule of 
Solution. 


Tuts Work appeared in the year 1744, about three 
years after the publication of his last memoir *, on the 
same subject. It was intended, and, with a few excep- 
tions, it must be conceded, to be a complete treatise; 


containing essentially all the requisite methods of solu- 


tion, with great abundance and variety of examples and 


illustrations. ‘There is wanting, however, to make it a 
perfect work, and on the subject, the best extant, a new 
algorithm ; a more compendious process of establishing 
the theorems; and certain supplemental formule, that 
determine, not the nature of the curve, if a curve be'the 
object of enquiry, but the conditions according to which 


jt must be drawn. ‘These desiderata were afterwards 


supplied by the fertile genius of Lagrange. 


The former memoir contained, as it has been already 
stated, abundance of valuable matter, but ill arranged. The 


* The volume of the Commentaries of Petersburg, in which this 
memoir is inserted, is said to be for the year 1736 ; which determines 
nothing concerning the dates in which the memoirs contained in it 
Were written. Several of them, however, were undoubtedly written 
after 1736; since they contain accounts of observations made in 1740, 


K 


66 Plan of Euler’s Work. - 


distribution and arrangement, however, of the present 
work is extremely luminous and regular. Absolute 
maxima and minima are first treated of, which concern 
curves that are to be determined solely by the property 
of maximum or minimum; such a curve is the brachysto- 
chrone*, which has the property of the least time, out of 
all curves whatever that can be drawn between two given 
points. The curve generating by its rotation round its 
axis the solid of least resistance, is another. 


If f V'dx be the analytical expression of the maximum 
or minimum, /” may contain either determinate or inde- 
terminate quantities, such as integrals. Euler first con- 
siders the former cases, that is, when V’ contains only 


sg dy @ fuigty 
quantities, such as 2, ¥, a = , &c. which are plainly 


determinate quantities, that is, of assignable value, when 


x or y 1s given. 


After absolute, relative maxima and minima are 
treated of: these relate to curves that are to be deter- 
mined not solely by the maximum property, but con- 
jointly by that and other properties. Such a curve Is 
the brachystochrone+, when the property of equal 


length becomes an additional condition ; that is, when 
the curve of quickestdescent is required, not amongst all 
curves whatever, that can be drawn between two given 
points, but only amongst those that are of a given length: 
such also is the brachystochrone, when a third condition, 
that of equal area, is added & | 


see esp ne soonest at ee sparen cat eles an EO a eS 


* See pp. 4, 9, 34. +t See p. 18. t See p. 47. 


Method of soloing Cases of absolute Maxima. 67 


In these cases of relative maxima and minima, the 
quantity V, when {Vdx represents a property, may, or 
may not, include integral expressions: and since, by an 
artifice like that which we have stated*, Euler re- 
duces all questions in which are involved two or more 
properties, analytically expressed by Vda, /¥ dz, 
{Xdx, to this form, 

SV dx+af¥d«+bfXdx+ &e. 
the determination of all cases is reduced, ultimately, to 


that of an absolute maximum or minimum. 


The method employed by Euler in treating the first 
and simplest cases of absolute maxima and minima, is 
similar to that used by him in the eighth volume of the 
Commentaries of Petersburg: thus, suppose the expression 


of the maximum property to be fda, and V to bea 


; d 
determinate function+ of 2, Wie , &c.; then, as it has 


appeared, the property belongs equally to the curve and 
to its element. Let PQR be a curve, and let its 


* P. 56. + P. 62, 


68 Mode of Computing the Variations, 

ordinates LP, MQ, NR, &c. be y’, y’,y”, &c. and 
Qq=1; now fV dx isto be a minimum or maximum; 
suppose f V..dx to be its value up to the ordinate y, the 
one immediately preceding LP; then, beyond that or- 


dinate, to the right of y, the value will be increased by — 


Vdxt+V' .dx+v".dx+&c.; so that, the whole value 
of fVdx will be 
JV .da+(VAVer+vV" + &c.) dx 

a maximum or minimum, and in which, for the reasons 
just assigned, 
(Vy +V'+ PV" + &e.) dx must be a maximum 
or minimum, that is, (dV+dV’+dvV"” + &c.} dx must 
be put equal to nothing *, when the changes arising in 
dV, dv’, &c. from the translation of the point Q to q, 
or from the increment or variation v are properly ex- 
pressed ; or, if dV=Mdx+Ndy+Pdp+Qdg, when 
the changes or variations of 

dx(Mdx + Ndy + Pdp + Qdq) [1] 

dx (M'dz+N dy +P dp +Q.dq7) [2] 

da (M"dx+ N’ .dy’+P"dp’ +Q’.dq") [3] 
are expressed. 

Now in the Figure p. 67. the ordinate y” is changed 


by the quantity », and since 
Mee We Arah a oy ee 


Bg. 2 ea aa de” 
“lle phar tt) f em Dt hap eae 
Kista «Gus mgy et ip apd ee 
SO i lal Mais ect sti ee 2 oi 
BET EK q dx dx m 

* See p. 53 


’ 


from the Differential Expressions for V. 69 


it is easy to see the change, or increment, or variation pro- 
duced in p, p’, &e. g, 7, &e. from thechange vin y”, whichis 
the sole ordinate that admits of any: thus, the variation * 
in p is =O; in eis ee an A Yea oe on ig sae DG 

Se cewiohinhiecist |S cidaadesty Poerrichac itive 0 a Bw 


is =. In a 18 — 
dx** 7 dx 


Hence, substituting in the three expressions [1], [2], 
[3], 
yee GD. v 
the variation in [1] is nies dx; 


ered (2) “is Cheat —) dx ; 


B- +> - in [3] is (V9 - PL + O55) de. 


Hence, the whole variation is 


thet 08 Q’—2’+Q " 
(NV rio gnt tea) dx.v=0; 
but, pases + = (the increment of P” being diminished 
In infinitum, ) 
Oe 2 Q’+ Q 
also ——+———_- = 
dx 
(Q4+2dQ+@Q)—-2(Q4+dQ)+Q_ dQ 
dx* ~ dat" 


_ * Variation, now-used technically, is not in this Tract used so by 
Euler, 


70 Resulting Formula of Solution. 


Hence, retaining, for the sake of homogeneity, quan- 


tities of the same order, 


dP dQ 
petganit dee Ve 


which formula will solve all questions of absolute maxima 
that do not involve integral expressions, or differential 


expressions, of a higher order than dq, or oY, and by 


means of the principle and formula stated in p. 56, will 
solve all questions of relative maxima that neither involve 


integral, nor differential expressions of a higher order 


2 


d*y 
dx?° 


than 


There are two points, in the preceding process, that 
require explanation. The increment or variation » is 
attributed not to the ordinate y ory’, but to y” ; and three 
values of dV’, such as Mda+Ndy+Pdp+ Qdq, only 
were taken. With regard to the first point, the value of 
SV dx, that is, SV da, from the left hand up to the 


ordinate y, is supposed to be determined, and not to be 


affected ‘by any change in that portion of the curve that 
lies to the right of y; now if ¥” be changed, this happens, 
or f V dx is not affected: for, an element of it to the 
left of y and immediately preceding y, would be, analo. 
gously to the mode of expression used, 

| Mdx«+ N.dy, +P,..dp.+ Q..dq, 

aad according to the table [a] p. 68, we should have 

ws Natio a 
Be Saar 
neither of which quantities depend on y”, the ordinate tha 


Explanation of Euler’s Method. wt 


is changed; and consequently, these quantities would 
suffer no change or variation, and therefore fVdx will 
not be affected. But, if to y' instead of y” the incre- 
ment » had been attributed, then g, would have been 


changed, and its variation would have been 


v 
Ax” 


This explanation includes, in fact, that of the second 
point: if a fourth horizontal series 


MM’. dv = N”. oT a hg dp” Ba Q”. dq” 


had been taken, then in each term, there would have 
been no variation from the variation » of the ordinate 


y”. For, according to the table [a] 


at? — a" a — 2yt 4 y : es 
a a a iN = deo Ea , which quantities are 


independent of y”. 


If the quantity V’ involve differentials of a higher 
order than d’y, the variation » must be attributed to an 
ordinate more remote from y than y’; and more hori- 
zontal rows like [1], [2], [3], p. 68, must be taken 
into the computation. We may easily learn from the 
preceding explanation, the exact process that must be 
instituted ; for instance, suppose 


[b] dVv=Mdxe+Ndy+Pdp+@Qdq+Rdr 
“dx da dx 
then, forming 7’, r”, &c. as p’, ’, q, ¢, &c. were 
formed in table [a] p. 68, we have 
% 


72 Formula of Solution when os enters into the Expression for dV. 


dx. dx i) 


y qg —q sheuts Oar 3y” ae y 
dx dx F 


T -F _y —3y"*+3y"-y' 


br 
I) 


dx dx . 
7 " ar or gid ray ir 3 yf” a ay — yf” 
dx dx’ } 


now, if » be attributed to y’, then the preceding value of — 


y -3Y +3y-y 
dx 


change in r; therefore, we must, instead of y”, make the 


ENE 


would be affected by the 


LA 


next ordinate, or y”, to vary by the quantity », and then 
7, and consequently, (Vda will not be affected by any 
change in the curve that lies to the right of y: again, 
four rows like to [6] p.71, are sufficient; for, if we 
introduce a fifth, since y’’, p’’, q*’, r’” do not involve y”, 
no change would be introduced in it, or its variation, 


arising from v, would be nothing. 


If the several variations in the four values of dV be. 
collected and reduced, after the manner given in p. 69, 
the result will be 

din GR 
Cite . "tae =e) de. 
which, by the nature of the question, that is, by the pro- 
perty of maximum, must equal nothing. 


The above mode of computing that peculiar incre 
ment of f Vdx, which arises from a change in the curve 
itself, and which Lagrange technically, and for disti 


¥ 


~ Rule for finding the Variation. 73 


tion, called the variation, is sufficiently plain and direct, 
The rule too that Kuler gives *, is only in different terms, 
the very rule now in use; for instance, to find the varia- 
tion of — y A (1 + p?) 

take, says he, the differential of this quantity as in the 
ordinary calculus; and then, instead of the differentials 
of y’, p, substitute those peculiar differentials, that is, 
variations, that arise from a change in one of the or- 
dinates (y’) ; accordingly, the common differential being 


di L+ d 
(dy) x V( P+ (ep) 
the peculiar differential, or variation, if (dy’)=v and 


: . . 
(dp) ae will be 


y J/ (1 +p?) +s Sat 


In the preceding formula, V contains definite quan- 
tities, such as x, y, dy, &c. but it may contain an inte- 
gral expression, or be of this form 7/2 dx; in such case, 
Euler employs a method similar to his preceding one; 
‘but, the method becomes very complicated: for then 
corresponding to V’ would be 7'f Zdx, and since the 
consecutive value of 7’ would be 7”, and of fZdz, 
f{Zdzr4-Z.dx, the quantity corresponding 

to FV’ would be T’f'Zdx + 7'.Z dx, 

eS T’f(Zdx + TT’ Zdx +7". Z'dx, 
and then from these expressions, d/’, dV’, &c. must be 
expressed, and their values put down, such as arise from 


the variation » of any ordinate as 2 y, 


ii A : 
* Methodus Inveniendi, p. 33, &e. 
J, 


74 Formule, when Integral Expressions are involved. 


It will be easily perceived from what is already said, 
that this process must be very tedious ; and it is not 
necessary more fully to explain it, since the method, 
with regard to its principle, contains nothing either novel — 
or abstruse, and as a method of computation has been — 
superseded by another more regular and concise. 


Since s= f'/(dx" + dy) =fdey/ [i+ (2) ] 
or since s is an integral, if 
dV = Mdx + Ndy + Lds, 
the formula for the peculiar differential or variation of — 
S¥ dx would be given by the preceding method ; which 
formula is to be adopted to the exclusion of those other — 
given (dV having the same expression) by Euler in- 


the eighth volume of the Petersburg Commentaries. , 


V has been supposed to contain an integral f Zd.,. 
and we may go on farther, and suppose Z also to contain 
an integral such as f Xd%, or V to contain a double inte- 


gral, and so on: the methods to be pursued for such cases 
will be similar to the preceding: they will, however, | 
from their complication shew how inconvenient Euler’s 
process is ; although, if we attend to its management by 
the author himself, we cannot help admiring the sin- 
gular art and dexterity with which, from a perplexed 
mass of symbols, he at length extricates compact and 
simple formulze of solution. 
| 

Euler in the sixth volume of the Commentaries of 
Petersburg reduced problems of relative maxima and > 


Relative Maxima reduced to Absolute. "5 


minima to a dependance on as many similar equations, 
as the properties proposed. For instance, if the curve 
required was to possess two properties, the equations 
would be of the form R.bg—S.ci*, in which S=R+ 
dR. But, Euler gives there no satisfactory nor general 
proof, that S=R+dR; but contents himself with 
saying, that from the analytical process, such will 
appear to be the value of S; and if S should not be 
under the proper form, he then directs the computist so 
to reduce the equation R.bg—S.ci, that the coefhicient 
analogous to S, shall equal R+dR. 


In the present Work, however, he gives a much more 
satisfactory proof of this important principle: the nature 
of which will be easily understood from one or two 
simple examples. Let } 


dV = Mdx + Ndy + Pdp, 


‘and let » be the variation of the ordinate y” and w that of 


y’; v and w» representing quantities such as bg and ci are 


in Bernoulli’s and Euler’s former method}; then since 


[see p.68. | 


M dd 5 ft 
AP die" AR UE ORNATE = 


ES pee, Ree es aide Le 


sae ® wee ya 
the variation of p = Tx? Ob ft Sa Ti of p” = —— 


yy P 
consequently, the variation of dV.dx = —.dx 


dx 


fh he  F TaEpelogau. 


“6 Proof of that Reduction, 


ee ‘ _ «nr y= w 
the variation of dV’.dx= (No ed. 7 de 
vail te oo 
exept galt, fa of dV’ .dx= (N v—P ~) de. 
Hence, the whole variation is [see pp. 55, 68. | 


(w 222) awe (wes SS yew, 


Now, WN” is the consecutive value of NN’, or = W’+dN’ 


P’—P’ is the consecutive value of P’— P, or=(P’—- P) 


: ee ew od 
+d(P'—P); consequently, if we make N’— 7 = 
° 7; i Leis Coase P 
R, the coefficient of the second term, or WW” — 7 


will be R+dR. Hence, the form of the resulting 
equation, will be 
Ro + (R +dR) Ve 

Again, suppose 

dV = Mdxr+ Ndy+Pdp+@Qdq; 
here, if the increments » and w are attributed to y” and 
y’, we must introduce an ordinate y, preceding y; since 
q, =2 th . Hence, referring to the values of p, q, 
given, p. 68, and operating as before; 


variation of p.=0, of p=, of y= ai , of p= 
| ic 


by Examples. 77 
Hence the variat". of dV,.dx = Q—— Or ..dx, 
| Nt” dae oO yv—2w ‘ 
baa 6F 4 of dV .dx (P5-+ 0a. ) da 
E77 7, (nD ,d¥— oO w— Qy | 
a .. of dV dv= (Not P. 2 —— ) de, 
” ma 1 ars 7 
Beals of dV". dx= (N"s Pionct 2 a 


or, collecting the terms affected with , and », the 


variation is 


eer Bea ey, dz + (NY—* 77 P's 
eres By, .dx; where it is plain, the coethicient of 


the term involving vdz, is consecutive to the coefficient 
of the term involving w.dx; or is R+dR, if the co- 
efficient of the first term is R; hence, as before, the 


resulting formula is 


Rio + (R+ dR)... 


The truth of this result may be also easily perceived 
from the form which the variations of p, p’, p’, and of 
q, 7, 7, &c. necessarily assume from their expressions 
in terms of the consecutive ordinates, y, y’, y”, &c. 


We may here again notice the important result, 
which Euler drew from the above formula. Since Rw+- 
(R+dk)s is derived from a property, of maximum or 
other, represented by {WV dx; if another property {Vdx 
be added, a similar equation, Qu+(Q+dQ)» may be 
derived : and exterminating », the resulting equation is 

R + aQ=0;; that is, 


78 Practical Convenience of Euler’s Formule of Solution. 


if Vv =Mdx + Ndy + Pdp + &c. 
and Y = w.dx+v.dy + 7.dp+ &c. 
dP ; dr 
r 7 as grime _-—— 
we shall] have R+aQ=(N 7 F&e.) ta(» 7, + &e.) 


= 
=(V+apv) ae + &e. ; 


that is, the same formula that would result, if the pro- 


perty proposed had been 
[¥ dx + afl’ Alas 


By this, all questions of relative are reduced to those of 
absolute maxima and minima: for, similar reasonings 
and properties hold, when the curve sought instead of 
two, has three, four, &c. properties: and if such pro- 
perties be expressed by 
dv ax, [i dxef Wada, Jf Uda, &e: 
then we must solve the question as one of absolute 
maximum and minimum; and inquire, what the 
curve is that has the property expressed by 
f[Vdxr+afF¥dr+bfWdx+cfUdz. 
Euler, besides the cases already mentioned, solves also 
those, in which V contains quantities, neither determi- 
nate such as 2, y, p, &c. nor integrals ; but expressed 


solely under the forms of differential equations. What we 
have given however, is sufficient to explain and illustrate 


Euler’s method. The results of that method are, for the © 


practical solution of problems, under a most convenient 
form. On that head there is nothing to desire. Neither 
is there any want of perspicuity in the principle or in the 
conduct of his method. It is the length of the opera- 


Tediousness of Processes establishing them. 79 


tions attendant on his method, the want of mechanism in 


his calculus, that are objectionable *. These inconveni- 


ences Lagrange removed: but, as in such eases it not 


unfrequently happens, whilst he rendered the process of 
calculation more expeditious, he deprived its principles of 


a considerable portion of their plainness and perspicuity. 


ene IS 


* Euler, we learn from the historical account prefixed to vol. X. 
of the Novi Comm. Petrop., was sensible of the inconveniences of his 
method : “Interim tamen ipsa methodus, etiamsi totum negotium satis 
expedite conficiat, tamen ipsi non satis naturalis est visa, propterea 
quod vis solutionis tota in consideratione elementorum curve investi- 
gandz erat posita, ista vero questio facile ita adornari possit, ut ex 
geometria penitus ad solam analysin puram revocetur, &c., 
autem auctor de hoc diu multumque esset meditatus, atque amicis hoc 
desiderium = aperuisset tamen gloria prime inventionis acutissimo 
Geometre Taurinensi Lagrange erat reservata, qui sola analysi usus 
eandem plane solutionem est adeptus, quam auctor ex considerati+ 
onibus geometricis elicuerat.”” Novi Comm. Petrop. tom. X. p12 


. Tametsi 


CELAP. Vile 


Lagrange’s Memoir — Use of an appropriate Symbol to denote the } 
Variation of a Quantity — Rules for finding the Variation —— New 
Process of deducing Euler’s Formula—Invention of new Formule. 


Ty the preceding account of Euler’s memoir, it can- 
not have escaped notice, that some ambiguity and much ~ 
circumlocution took place in describing the process of f 
finding that peculiar increment which depends on the 
quantity »v, by which the ordinate y or 7’ is increased. 


These inconveniences Lagrange obviated by a simple 
invention; that of a symbol such as d* ; which, analo- 
gously to the symbol d of the differential calculus, was_ 
to be the means of representing either a quantity or an 
operation. ‘Thus, the quantities » and w, by which y 
and 7’ are increased, are symbolically denoted by dy, dy’: 
and asd in d (ay"+6y’”) signifies, that the operation 
of taking the differential of ay”+ by’ is to be made; so, 
din 3d (ay”+by”), signifies that the peculiar differential — 
of ay"+by™ is to be taken, when instead of dy, dy’; »_ 
and w, or dy, dy’, are to be used. 


* ‘* Mais avant tout je dois avertir que comme cette methode exige 
que les mémes quantités varient de deux manieéres differentes, pour ne— 
pas canfondre ces variations j’ai introduit dans mes calculs une nouvelle 
characteristique 0.7? Miscell. Taurin. tom, II. p. 174. 


Similarity between the Differential Calculus and that of Variations. 81 


) 


_ With this change in the system of symbols, and with. 
certain changes in the processes of establishing the fun- 
damental formule, Lagrange called his method, a new 
method of determining the maxima and minima of inde- 
finite integrals. But Euler, resuming the subject in the 
10th volume of the New Commentaries of Petersburg, 
p- 54, called it by a name which still adheres to it, 
The Calculus of Variations. 


The variation of ay” + by” is nay"™.dy +mby” dy’. 
For if we recur to the problems solved by Bernoulli and 
Euler *, we shall find the difference of two contiguous - 
States of a quantity at its maximum or minimum put 
equal to nothing ; not indeed the whole difference, but 
the first term of the difference ; for the terms involving 
(bn)’, (co)’, &e. + are neglected. Now, the difference in 


that example, is : ; Pb ory being 


SM ie ts eae 
/ Pb V(Pb+bn)’ 
increased by bn: if y had been increased by dy, the first 


term of 


Jia Tinian would have been the 


common differential; consequently the peculiar differ- 
ential, that which is now to be called the variation, differs 
from the common differential, only inasmuch as bn or s 
or dy takes the place of dy. The process then for 
finding the coefficient of the term involving bn or dy, is 


© Pp, 335: &c, + P. 19. 


M 


precisely the same as that for finding the coefficient of — 


82 Rules for finding the Variations of Quaniities. 


the term involving dy; that is, for finding, what tech- — 
nically* is called the differential coefficient. The rules ( 
must consequently be the same: in other words, the | 
rules of the differential caleulus, become, with the " 
alteration pointed out, those of the calculus of variations. 


Hence d(y”)sny".dy, which is the first term of 
[(y+dy)"— y"'] expanded. 
Again, 0 (ay* + by*)* = + (ay + by*)-*. (2ay +4by%) dy. 


If dx be the variation of 2; then since the - 
differential of xy, or d(xy) is=ady + yda, 
the variation of xy, or d (xy) is=axdy + yd. 


If we take the example, p. 73, 


since d[y../(1 + p’)] = =dy /O+P)+ Faas 4 P 


sy /+p’)] =%%V0 ++ aay? p 


, 4 
y—y_ dy. . 
In p- 68. = = tae; 
hye het UI idyy dy’—sy _ ddy 
BU Dee pepemcimiy | Mase 
dy ddy 


od: 
Hence, Jp =" or=—~. 


If F=ax* /(1 + y’*) hyp. log. px e* 
|e = number whose hyp. log. =1 A 


* Princ, Anal, Calc. p.74. if 


- 


Examples to preceding Rules. 83 


ax>y hip.en 
v (4x) 


Baa </ (1 ty) 7 erdp+aa'/ (1 +y’)h.l.p.e*m.dq 


therefore putting MW, N, P, Q, for the coefficients of the 
terms involving dx, dy, dp, dq, we have 
dV=M.sx+ Nodyt+P.dp+Q.dq, or expressing M, N, 
P, Q, as partial differential coefficients *, 


dV dV dV Ww 
Va, eee hay dy! Mae qt. 


then since dV =3ax*V/ (1+y’)h.1. p.e".dx + — dy 


Hence, generally, whatever be the function /, 
if dV=Mdx«+Ndy+Pdp+Qdq+c. 
then dV =Modx+ Noy + Pdp + Qdq4+&c. 


Since the processes for finding the differential and 
variation differ only in the symbols dy, dy, which are 
arbitrary; it is plain, if both operations are to be per- 


formed on an analytical expression, that it is matter of 


* Princ. Anal, Calc. p. 79. 

+ This rule is not only in the first solutions of Isoperimetrical pro- 
blems, virtually acted upon, but expressed. “Il faut bien remarquer 
gue la difference des fonctions de deux lignes comme RO, RT 
(y,y+9y) qui se surpassent d’une quantité TO infinitement petite 
du second genre, se trouve en differentiant simplement la fonction de 
RO, et en multipliant par TO (dy) ce qui en vient, ayant omis les 
differentielles. Par exemple, si RL (Y) fonction de RO (y) etoit 
seulement la puissance z de la méme RO (y), en quoi consiste le cas 
de mon frére, c’est a dire, que si la courbe BH etoit une parabole du 
Jegré n, alors LM (3Y) ou RO*X—RT” [(y+dy) —y”] seroit=n. 
RO x TO (ny’—!.dy).” John Bernoulli, Acad. des Sciences, 
1706, p. 236; also Opera, tom. I. p. 424; and Euler, Methodus 
Inveniendi, p. 33, &c. gives the very rule and method for finding the 
variation, which Lagrange invested with appropriate symbols, 


84 - Order of the Symbols d, 3, changeable at Pleasure: 


indifference, which operation is performed first : or, if the 
symbols d, 3, meet together denoting operations, we may, 
at our pleasure, change their order: for instance ddy 
and ddy are alike significant; for ddy means the first 
term of two successive values of dy, or=:0 (y-+-dy)—dy 
=dy+ddy—dy=ddy: again, if V for instance be a- 
function of y; then 


dV ad’V 
ems and Woe esc as | 


dV PV 
riage and at maya ate 


7a bey, 
or, in a particular instance, when V =y”, 
d (y") = 1* term of [(y+dy)"—y"] = ny"—.dy 
dd(y") = 1* termof ndy x [(y+dy)"-y"" | = 
n(n—1)y”~.dy.dy, | 
d(y") = 1 term of [(y+dy)"—y"] = ny” .dy 
d3(y") = 1* term of ndy x [(y+dy)""-y""] = 
n(n—1)y"*.dy.dy ; 
. dd (y”) = dd (y"). 


And, by similar processes, d°3V' = dd*V = dddV = dsdV 
dV =3d'V = ddeV = bddV, 


This rule is, in Lagrange’s method, of the greatest: : 
importance ; it 1s an essential part of it. Amongst other ) 
uses, it enables us when an integral is concerned, to 
introduce the symbol é within the symbol (f°) of the 
- Integral: thus, since the symbols d and /’ indicate re- 
verse operations, 


_ 


. also of the Symbols 3, f. 85 
V=dfV; +. = ddfV= [by Rule, p. 84.] dd fV. 


Hence, taking the integrals on each side 


fs =fOfVissfV... [a] 


This result may be easily extended to double and 
treble integrals: for if V=f{/W, then W=dfW=f i 
by [a]; 

o [Vaffe: but [IV =dfV=d fl f[W, consequently 
ff W =f fev. 

Before we proceed to explain the last improvement 
made by Lagrange, we will give, under the symbols of 
his new Algorithm, another solution of the brachysto- 


chrone, [see pp. 4, 9. | 
Let OTS EB icaay, CL=ds, LD=ds, OA=z2, 
OB=2', g=324 feet; then the velocity = V 2¢2, 


Oo H 


A 

B 

= D 
hence ds + ENEMIES minimum ; 

Re, ont) tnt ee a a 
ds ds’ dds dds’ 

- a —~+ ——=090 
2805, + Fy) = 0, or 4 Fy : 


but ds= / (da*+dy’); .. dds Higa ee and dds’ = 


86 Brachystochrone solved in the Symbols of the Calc. of Varias. 
dy! .ddz’ dy.ddy | dy'.ddi/ 
EGAN, 1eh Ce, 7 1s Sea an 
ea hs, 
.. since KD and AC are not changed in the trans- 
lation of L to G, dy +dy 1s constant; .. ddy= —ddy/’, 
dy dy! 


=O [a]. But 


/ 


as 


and substituting in[@], 1.1. 7 | rae hes a constant ~ 


quantity, the property of the cycloid. 


This solution is on the same principle, as the first 
and second, but is less peculiar and geometrical than 
either ; instead of the similar triangles in the second, 
one of the processes of the new method above described, 
has been used. 


The rules in pp. 84, 85, relate to the second improve- 
ment made by Lagrange in thecalculus of variations which 
we shall now describe. If we refer to the method which 


Euler used for deducing the formula, NV _ + &c., we 


shall perceive that its length and complication arises 
from the integral {dx being broken down into parcels 
and elements Vdx, V’dx, &c. and from the calculation of 
the variations in the differential expressions that represent 


dV, dv’, dV", &c. [pp. 68. 69. | 


Lagrange precluded the necessity of this resolution 
of f dx into its elements, by combining with the 
preceding variation processes, an integral process. This 
will be understood from the solution of the following 
problem, in which it is required to find an expression for 


df Vdx, V being a function of a, y, p, g,7, &e. the law of 


— — .. 


ee 


Formula for the Variation of f Vdx. 87 


the formation of p, g, r, &c. being p = 2 nies ee Me =, 
&e. 
df{Vidr=fi(Vdz) [p.85.13.]=foV .de+f Vdd. 
Now on the principle of this formula, fady=axy—fydz 
SV dix.= Vide — fdV dx. 


- Hence, 3fVde=V du+-((dV .dx—dV 3x) 
Now since IV =Mdx+ Ndy+ Pdp+Qdq+&c. 
if dV =Mdx+ Ndy+ Pdp + Qdq+ &e. 
Vv .dx—dV dx=N(by.dx—dydx)+P(sp.dx—dp.dx) 
+Q (dgdx—dqdx)+k&e. 
= Ndx (dy — iyo + Pdx(dp— qex) 


+Qdx(ég-—rdx)+&ce. (since p= + ly tae =, &c.). 
ou, ddy dz cree 
Now p= Wepe =e oe em — (dey — p.ddx) ; 


+, dp —gax = (diy —pddx— qox .dx) = 


1 1 
ai(dey - pddx —dp.dx) = 7,4 (dy — psx) 


Now since the quantities p, g, r, &c. are formed after the 
same law, the same relation that subsists between dp — qdx 
and dy — pdx, must subsist between dg —rdx and dp — qex, 
and so on; but the relation between the two former 
quantities is thus expressed : 


1 ; 
dp— qox = 7d (ey — pdx) 
+. 8g—rdx=7 d(dp—qsr) =* 7>.d" (by p32) 


LLL LLL LL OC CC A AE A tC 


. ‘ ] ] 
% daz b 71g = —, — "4 ¥ 
eing constant, otherwise pm d(dy—pdx) 


88 Formula for the Variation of fV d x. 


and dr ~ sdv=7-d(éq—rde) = (dy—pdx). 


Hence, for the purpose of abridgment, putting 


dy—pdx=dw, we have, 
Pddw Qd*dw 
df Vdx=V idx +fdx (Nida + —— ei wre 


But by Formula, p. S7 sels Bs 
[ Pddo=P.dvo—fdP.do, 
fQd da = Qddw ~f[dQddw = Qddw— dQ dot fdQde 
S Rdda= Rd do—fdRd'so 
= Kddo-dRddot+felR.ddu, 
= R.d’de—-dR.dda+@Rio— {OR .da. 


Hence, collecting quantities involving like symbols, 


dP yee BR 


(Vda =Vdx+ fdvdu( N- = Bi pete 


=~; +.&c.) * 
[4] WR Ge Se ie eae 


3 (Q- ot + &e.) dw 
+e. + K. 


The quantity K represents the sum of the corrections — 


introduced by the integrations. 


~ 


, aP @&2Q 


a: I &c. are not partial differential coefficients, but the first, 


second, &c. entire differentials of P, 2, &e. me by oe he dihn e 
AQ, Ke. 


respectively. They ought in strictness, to be written = dP, = =. 


- 
EE a ee ee 


Formula when the Variation is taken between two Limits, 89 


If x be supposed to have no variation, or if dx=0, 
then dw=dy—pdx=dy; and consequently we shall have 
the variation of df Vdx by omitting in the preceding form 


Vsx, and writing dy, dy, &c. instead of dw, ddw, &c. 


The preceding value of the variation may be sup- 
posed to be taken between two limits corresponding to 
values a, and b, of x. Let y,, P,, Q,, &c. be the 
values of y, P, Q, &c. at the first limit when r=a; and 
my, 2 ,., 4... ac. when x=5; 


then, ¢fVdxe=V,.d2, — V,.dx, 


+ fda du (N- <4 a eee) 
a (P,- H+ &e.) OW, 

A ~ a. Tee + &e.) O05 
+ &e. 


If at the two limits a=a, x=b, the values of y,, y, 

are given, then dw,, dw, are both equal to nothing, 

(since d#=dy— pdx); and in this case the variation of 

[¥V dz is reduced to the quantity under the integral 

sign (f), which, in the case of fVdxr=a maximum, 
since 0f Vdx=0, must equal nothing; or } 


APP Q oi Rel 
a Te a Pa ee tah 


N 


This last formula, is that which Euler, by the method 


N 


90 ) Definite and Indefinite Parts of the Formula. 


described in the last Chapter, arrived at*; and it is here 


deduced on his hypothesis of the evanescence of dy,,. 


dy,, &c. 
The formula V sere &e. is not, however, 


equal 0, solely in the case when dx,, dx,, dy,, dy,, are 
equal 0, but also, when these are, from certain equations, 
assignable quantities. For, the formula [4a] p. 89, is 
composed of two parts: one, affected by the integral sign, 
expresses the sum of all the separate variations throughout 


the whole extent of the curve or integrated quantity ; 


i il ———— 


the other part, independant of the integral sign, is | 


affected only by the variations at the extreme points, and ; 


therefore cannot by any combination with the other, 


(which by changing dx and dy may be varied at will) © 


form a sum equal to nothing. Hence, since dfV da _ 


must =0; each part separately, the one under the inte- 
gral sign f, the other not affected by it, must =0. 


M. Lagrange is the inventor of that part of the | 


* Methodus Inveniendi, &c. Prob. V. Lagrange, Miscell. Taurin. — 


tom. II., says, “‘ Mais les formules de cet auteur (Euler) sont moins ge- 
nerales que les notres 1° &c.—2°, parcequ’il suppose que le premier et le 
dernier point de la courbe sont fixés, &c.” Euler also, in his memoir 
subsequent to Lagrange’s [ Novi. Comm. tom X. p. 110.] acknowledges 


that his first formula did not contain the absolute or definite parts in-_ 


volving dw, dw, &c. ‘* Neque tamen he partes absolute frustra sunt 
invente, sed singularem prebent usum, ad quem methodus mea prior, 


, oP sale Sass 
quz tantum wquationem Nr cet: &c.=0 suppeditavit, minus est ac- 


cemmodata ; quam ob causam hee methodus illi longe est anteferenda.” 


J 


Lagrange’s second Problem. 91 


general formula which is not affected by the integral 
sign. Its use, as it will be shewn hereafter, is great and 
extensive: without it, the solution of problems would be 
incomplete ; for, Euler’s formula, that under the integral 


sign, merely determines in general terms the relation of 


x and y; undoubtedly the chief, but not the sole object 
of search. 


We will now proceed to that, which, in fact, is 


| . ; 
Lagrange’s second problem*. 


It is required to find an expression for f/Vdx, in 


which dV = Td ft, dt not being similar to dy, dp, dq, &c., 


but determined by this equation ¢=fZdz. 
By p. 87, of Vdx = Vbx+f(dV.dx — dV dx) .... (1) 
but V=T.st; «. Wdx-dV dx=Tdx.dt—Tdx.dt. 


again, by [1] dt=0fZda=Z.dut+f(8Z.dx—dZ sx) : 


- Tdx.dt — Tdx.dt=TZdrdx—TZ.dxdx + 


Taz Sf (@Zdx — dZ se), 


=Tdx f(dZdx-dZ.dx) 
d 2. df VdeaVba+fTdu {(sZda—dZsx). 


Now by virtue of this formula /(vdu) =vu—/f (ud), 


if h be put=/T'dz, the latter part of the value of df Vdx 
is, hf(éZdx-—dZdx) — fh (sZdx—dZdx) 
Also by Form [A], p. 88. 1fdZ = Mdx+Ndy+Pdp+&c. 


dP d 
hf(eZdx—dZdx)= hfdx de (Make H+ GE be.) + 


er, | 22 | 


* Miscellanea Taurinensia, tom. MI, p. 183. 


92 Lagrange’s second Problem. 
and fh(dZdx—dZdx) = 
d(hQ) 
dx* 


fax. doGN- 
(L, L’ being put for the definite parts,) 


= &e.) ly oy Ja [3] 


Subtracting this from [2] and adding V dx, we shall 


have the value of d/Vdz. 


In general, it is required to find the whole variation of - 


fV dx from «£=0, for instance, to =a. In such case, let © 


the whole integral of T’d«#= H; then the part [2}, 0 of the 


variation = Hfdaxdo( a crs Q 
ee OT) | ke.) + HL 


For since f/isconstant, Hf N= fEN, and H.dP =d( HP); 


subtracting therefore, as before, [3 | from [2], we have 


df(Vdx=Vex 
ee d(H - DB, d(H-h)Q . 
+ fdx.do((H-h)N— ae ke.) 
d(H—h)Q 
[B} wee one h)P “Ae +80.) a 
kc, 


If we put H—h, thatis, H—/{Tdx=k, and suppose. 
dx, dw, the variations at the limits, to equal nothing, we 
shall have 


oY Vde=f deta (NSO EO) _ i.) 


Application of his Formula to particular Cases. 03 


which is Euler’s formula, Method. Inveniendi, &c. p. 91, 
and the subject of his third Chapter. 


If dV=yp.dx+v.dy+m.dp+&c.+ Td, 
which is Lagrange’s case. Misc. Taur. tom. II. p. 183. 


dfVdxe= Viet fade .dw(s —“4&e.) ip ik 4 


+f dedo (kN-— sae + &c.) 


“3, 
S being put for the sum of the definite terms. 


If dt=ds=V (dx*+dy’) and dV=pdxr+vdy+Tds 


d( | H— {Tdx\—-—~ 
af Ve = fdxde|»— Scolebindessez 2) 


not taking account of the definite parts ; 
For, dt=ds=dr// (1 +p’ )=Zda; .*.Z=,/(1+p*) and 


aL =— .dp ; which, compared with the value’ of 


_dZ [p.91,] gives N =o, j 2 MASE Sd elt hee 


VW (4p)? Q=0, &e. 


Hence, in the case of maximum or minimum, since 
df Vdx=0, there results this equation of solution, 


Be 26 Fe aay yy Py LS 
yd - d(H J Tax\ Fp) = 93 
whereas Euler, Novi Comm. tom. VI. p.141, gives this 


| eee ETE 
TS OEPD 


94 e Lagrange’s third Problem. 


y answering to M, T to L, and aes being = ¢ 
nie 
8 


This is the case before alluded to [p.61.|; when it 
was remarked that Euler’s formule contained in the 
sixth and eighth volume of the Petersburg Commentaries, 
were erroneous, when / contained an integral ;_ which 


integral in the above instance is s. 


In the third problem of Lagrange [p.185.], it is re- 
quired to find the variation of f/V¥ when V is given 
simply by a differential equation involving no differentials 
of V higher than the first. 


Let dV — X.dx+ Udx =0 be the differential 
equation, X being a function of 2, y, p, g, &e. and U 
a function of a, y, p, g, &c. and of V’; then taking the 
ee and supposing da to be constant, 


wy - (Sade Hagler) ie 
+ (Gade +7 +a <dp +e.) de 
dU 
tap: W dx =0, 


or d*’V — do.dx + Vax ie 3 3 eae 


Substituting d@ for the collection of terms involving dz, — 
dy, dp, &c. Hence, if instead of deducing the diffe- “ 
rential in the last operation, we suppose the variation to — 


be deduced, 
dW - t9.de4THde20 [r= 48 


Lagrange’s third Problem. 95 
multiply by a, and a d3V— (ado) dv + (a T.38V). dx =0, 


or, d(AdV’) —da.dV = (xd) du+(aT.V).dx=o0... [0b] 


Assume the sum of the second and fourth terms to <0 ; 
ane : ; 
“. dA—aT.dx=0 ; and >= Td «x, and integrating a= e/™ 


[e=number whose hyp. log. = 1.] 


Hence substituting this value of ain the sum of the 


first and third terms, 


Sabine f Gola One 
CRITE. CBEST eA fess Oda 


Euler, by a different process, solves this problem in 
the third Chapter of his treatise on Isoperimetricals. 


Euler, however, stopped at this problem; and did 


not, as Lagrange has done, proceed on to those which 


besides dV’, involve d*V’, dV, &c. Suppose U to be a 


function of x, y, p, &c., V and of ae then in taking the 


da’ 


differential [a] p. 94. of the equation dV — Xda} Udz =0, 


/ hf 
an additional term a VY € ses oa) will be introduced, 
dy ax 


a : dU 
and consequently in the variation, this term ia dV; 
v 


therefore when the equation is multiphed by a (for the 


same process must be used,) there will be an additional 


term AT’. dBV (T! =<) =d (xT! AP) ~ d(aT’) x oV; 


: . equation [6] 1.2, would be 


06 Third Problem made more general. 


d(a9V) —da. dV —aso.dx+(aT.sV) du+d (aT’.dV) — 
d(aT’) 3. 


Hence, making the sum of the 2°, 4", and 6" terms =0, 
- dx+a.Tdx —-d(aT’)=0...[c] 

and d[(a+ aZT")dV |] —- asodr =0.... [d] 

in which two equations, a must be deduced from the first 

and substituted in the second. 


‘The process is the same if U besides a function of 
the former quantities, is also a function of d*V ; for then in 
the equation of variation there would be introduced a 

dV 


term T'". dd (Fr => w=) ; and consequently 
@&) 


in the equation multiplied by a, a term = aT”. d’sV" Ay 
d(aT"ddV’) —d(aT”) .diV=d(aT". dev) —d*(aT” aV) 
+ aT” x dV. 


Hence the equation [c] 1.4. will become 
— dra +a.Tdx — d(aT’) + @.(aT)=0, 
and the equation [d] 1.5. | 
d [(ataT’— d(aT’)) ]—d(aT’.dsV) —rdode=0: | 
and similarly, if differentials of a still higher order are 


introduced. 


In the preceding cases, the maxima and minima — 
depend ona function of one variable quantity alone : but q 
a variable quantity z may be introduced, a function of — 
two others.x andy. For instance, suppose it were requir-_ 
ed to find amongst equal solids, that which is bounded — 
by the least surface, (which is the problem given by 


Problem of the Solid of least Surface amongst equal Solids. 97 


Lagrange in his first Appendix, p.188.): then, if the 
points of the surface be referred to. three rectangular 
co-ordinates x, y, z, we have 


apatiae! 

My brine the parte differential soatctents * 
=)» ve eing the partial differential coefficients *, 
we have also the solidity =//f'z.da.dy, and the surface = 


[far.iy JT +) + Gy 


the integrals being taken, first relativelyto x, then relatively 
to y. The problem therefore analytically expressed is 


[a]. «nv ffd(z.dx.dy) 


+ af fo.(dx.dy)A/ [1 =e (@)+G)] 45) 
now VL +) + G)] = 
a(n) * eG) +G + G 1. 
a. ten (Es am +] =? 
© L+G)+G@)T"=8 


dz OP as 

then (P17) = SP Gs = PGS ae 
By : P) dQ 

similarly af ai(F) rer: = @ 5 iret: 


eS . * Princ. Anal. Cale, p. 79, 
0 


08: Problem of the Solid of least Surface, SC r 


Hence the equation [a] p.97. 1. 10. becomes 
ffdu.dy.dz+afPdy.dz+ af Qdx.dz 


dQ 
~ af fdv.dy — sz—affdx.dy Tt =o. 


’ 
eS. lO 


Hence, making the sum of the second and third 
terms, which are affected with one sign of integration, 
and belong to the extreme points of the curve surface, = 

Pdy+Qdx=0, and consequently the her 


ters = O:yor: 1a. — 0.5 = 0, [p] . 
now the condition under which | 
Mdx+ Ndy 
is a complete differential, is this * 
dM dN 
dy ~ da 


Hence, putting N=xc-—aP, M=a®, we have 
dQ dP dP dQ 


a——-=1—da.=—, or, | — a—— — a=— =0,7 
0 


DE AG 2), 
which is the equation [p] 1. 8.; hence, (« ~a@P) dy + 
dz 


aQdvxisa complete differen’.: butthat, dz Age +——dy 
{ - dx dy “4 


may be real, Gere dy must be also a complete 


d* i 
differential; that is, must=+—~-. These two: 


ad’ 
dx.dy dy.dx 


* Woodhouse, Princ. Anal. Cale. p. 88: also Euler, Cale. Int. p. 3155) 


Portion of a Sphere satisfies the Conditions of the Problem. 99 
conditions therefore must be fulfilled, in order that the 


problem may be solved. 


If r be the radius of a sphere, and «, @, y, the three 
co-ordinates of its center, 


r=/[(e2 — a)? + (y — 6) + (« -»)']; 
ds = oo dat(y-@)dy]; 


& awe as _ 
.. x—«), and —~ 
Be. ty: stein Tey 4, fies 
. (putting for z—y its value) ia i Pe one con- 


dition therefore is satisfied: 
=F. “.(« — aP) dy +aQdzr= 


( -“) ady +- o yde + dy — TF dx: which is a com- 


‘again P= 


_; .*. the portion of 


| plete differential, making 1 — i gaa 
Te 


ithe sphere corresponding to x, y, 2, satisfies the problem. 


The formule of solution, on which the nature of the 
“curve, or the relation of x and y, depends, were all, .ex- 
cepting the two last, invented by Euler. To Lagrange 
‘belongs the merit of having deduced them by neater 
processes. ‘The latter author, -however, is the sole 
inventor of those definite and absolute formule [see p. 88. 
1.13, 14.] which are requisite for the complete solution 


of Isoperimetrical problems. 


_ Inhis first memoir, Lagrange seems not exactly to 
have comprehended the nature of these definite formule. 


100 


He drew some conclusions, which the Chevalier Borda * 
proved to be not general. These defects, however, 
M. Lagrange remedied in a subsequent memoir in the 
fourth volume of the Miscellanea ‘Taurinensia: and 
without acknowledging the detection of the defects, or 
rather, with a faint endeavour of denying it, extended 
his formule, more accurately applied them, but con= 
firmed the truth of M. Borda’s results. 


In the same memoir Lagrange considers the subject 
under a new point of view, and gives a method of solu- 


tion including all his former ones. 


The substance of Lagrange’s first researches on this | 
subject have been given, with some deviation, not essen-- 
tial, from their mode. -Euler’s last manner + of treating 
the subject has been followed. ‘That is commended and 
adopted by Lagrange in the latest of his publications ¢. 
In the same Tract he has resumed the consideration of 
the general problem, and deduced formule applicable 
to all cases. 


This last method of Lagrange’s is distinguished 
rather by its mode of treating the question, than by any 
thing novel in its principles; and therefore, it will be 
separately considered in the following Chapter, which 
the Student who hastens towards the end of the j pie 


may, without inconvenience, pass over. 


Dr = ee aS, o> oh een a ice 1 


* Acad. des Sciences, 1767. + Novi Comm. Petrop. tom. X, p- 514 
t Legons sur le Calcul des Fonctions. 


CobiAer.. V1. 


Lagrange’s general Method of treating Isoperimetrical Problems— 
Equation of Limits—Cases of relative Maxima and Minima re- 
duced to those of Absolute. 


Ler V be a function of x, y, p, 9g, 2% p’, q’, &e. 
zz _dy iui Te Adan 28 5 
caine pas sia = Pw ~ dx’ » gee: ihe. 
and let dV = Mdx + Ndy + Pdp + Qdq + &c. 

+v.dz + mdp'+o.dq' + &c. 


Bioiy tera; EN ah a. 
dx ax 


Y= P _ dQ + &e. 
dx 
y’= Q — &e. 
&e.= &c. 
dr de 
rh ae ae ee ree e — &e. 


dr 
yA = c= ae a. «xe. 
Be ce. 
iC. — Rhee. ' 
we shall have, by processesalreadydescribed, | pp.87, 88. | 
i{Vdu =Vbx+ fdudo¥ + fdaddZ 
+ ¥’.d0 + Zidw 
+ ¥".ddo+ Z’.ddw 
+ &e. 


102 Lagrange’s general Method. 
Hence, in the case of amaximum, when dfVda= 
Vi30+ Z.dw = 0, 
or Foy + Zdx =0, if dx = 0, since [p. 88. | 
dw = dy — pox, and dw’ = dz — pda 


If y, x, are quantities independent of each other, 
then,s V=i0;.and “Z'°= 0, 


But, if the quantities y and z are connected together 
by an equation, such as f (a, y, z) = 0, or W=0, 


then, supposing x invariable, we have 


ams dt ae Re); 
which combined with the former equation [1]. 3.] gives 
dy CL 
Le rae LT —— 


The same result will be obtained, if x be supposed 
to vary; for then 


dW dW dy 
ae I te hai dz = 0, 


which combined with the common differential equation 


Le dy Le 
Tt adh ae -- Pi ae 0, 


ihn 14 ole dy dz 
GIVES, dye (dy ~ da tar 1-758) = 0% 
dW 


dy” 
which combined with Pde + Zdw’, gives 


or, 


60 -{- ee dw =O, 


Lagrange’s general Method. 103 
dW | 
— — sdb 0, as before. 
dz d y 


Hence, this conclusion follows; that the variation of 
# in no wise affects the general equation of maximum or 
minimum, but solely the equation at the limits. 


The equation at the limits depends on the parts of 
the formula for df Vdzx, which are definite and freed 
from the integral sign, and is 

08 — Ader, 
+ ¥’,.d0, + Z,.dw, + &e. 
=OF'S dus Ze Swish bel 
V,V 5, %x,,%x,, &c. representing the values of V, 32, 
&c. at the end and beginning of the integral. 


Instead of eliminating dw, dw’, by the combination of 
the two equations [p. 102. |. 19, 20.] we may multiply 
the latter by an indeterminate quantity a, add it to the 
first equation, and then determine the resulting equa- 
tion, by the elimination of a; thus 


: : dy dW i, 
Videt Z.do + eT + Aa ou = O. 
Eliminate a from the two equations 
dW dW 


0, 


ua and Zz a say Mure 


and there results, as before, 


aly dy 
ale om = me = Oj 


_ which combined with W=0, will give the values of y 
_ and z in terms of z. 


ie Pe ae 


104 Lagrange’s general Method. 


This last method extended, will afford a general for- 
mula of solution ; for, suppose to be a function, not of 
x, y, %, only, but also of the differentials of these quan- 
tities ; that is, suppose 

dW = M'.dx +N’.dy + P.dp +Q.dq +&c. 

+ v.dz -+- n dp i o.dg + &c. 


sad ee @y'aAwe4OPY ,o0m) ys 
ee Vie Ale / d(rQ’) aS 
[2 jose At o rf + &e. 
&c. 
Se eat ee) 
(B # ays atte 


CA. Melt + &c. 
&e. 
then we shall have 
dfVdxet+dfaw).dxe=(V+awWw) sx 
+ S([¥4 (1). dx to] +f [Z+ (Z) dx du’ 
+ [P+ (P)]u4 [2 + (Z)] oo 
+ &e. 


In this formula since W=0, aW. dx =0, andsince dw, 
dw’ are independent quantities, we have, on principles such 
as have been already stated, > 

) oa © es ty Z+ (Z) =9, 
whence A is to be eliminated ; and then by the aid of the — 
equation of condition #”=0, y and x are to be deter- 


mined in terms of x. 


This method comprehends all the former ones which 


Lagrange’s general Method, 105 


were separately instituted [pp. 88, 92.] to determine the 
* variation of fV dz. 


Ist. When dV contains merely 2, y, p, q, &c. and 
_no relation is assigned between 2, y, %, then the equation 


W=0 is not introduced, and the formula becomes 
Y=0, Z=0; 


9 agent he) 
or V — Ena vee — &c. = 0, 
amd » - 5 oo Be. = 0. 


2dly, If V contains an integral expression; that is, if 
dV = Mdx + Ndy + Pdp + &c. + Tdt, 
in which ¢=/Sdz, then the equation of condition W=0 


becomes dé — S.dx=0, and consequently, / ee _ S) dx 


=a. Hence, substituting ¢ for z, we have 


AP) dtr 


Y= Ne tae a Vass — &ec. 
Z=T [since »= 7, =0, &c.] 
ie) aig cued Aa) RS 


[af dS=M'.dx + N’.dy+&c. | 
ox 35 [since  =0, 7 =1, o°=0, &c.] 


_ hence we have from the equation Z + (Z) = 0. 
T - = Lighthence? 2S PULL Pda — HT, 
if f Tdx =H, when 4=0: substituting this value of A in 


the equation Y + (Y') =0, there results 
P 


106 TLagrange’s general Method. | 
dP 20 aH =fTav)P) 
N-& +5 2~ he. + (H-fTde)W AE 


+ &c. as before, p. 92. 


3dly, If dV —@.dx+U.dx=0, and Uisa function of : 


V,, this answers to the equation /”=0, and W= Jf —@Q 


y 
+ U,and dW=T.dV + had — dQ; instead of z, consider — 
V’ to be the variable quantity, then in 
dW=do+v.dz+.dp+&e. Y=T, r = 1, 
the other coefficients being 0 ; 


Hence, Y = 0, (Y) =aN- 


_ + &e. 
dx 


*, Z+(Z)=0, or aT'— a tie x= e/™ as before, 


dx 
p- 95. 


The same method applies to equations in which 
higher differentials than the first are involved; for in- 
stance, to equations such as 


yp De eer Na | 
Mae Sade 7 ee 


In the variation of (Vda, the parts under the in- 
tegral sign, that is, ¥+ (I), and Z+(Z) equal na ; 
there remains therefore, supposing dv = 0, 

Vide = (¥' + (P)]ey + [Z + (Z)] a 
+ [F+W \]dby + (27 + (2)]dae 
+ &c. 
which expression must be substituted in the equation of 
limits, that is, in 


! 
v 


Lagrange’s generul Method. 107 


dU, — dU, = O, 
putting U= fVdx. 


At the limits, there may be particular relations 
between x and y; for instance, it may be required to 
draw the brachystochrone between certain curves. Let 
L=0 be the equation to the curve at the first point, and 
M=0 at the last point, that is, let L be a function of x,, 
-¥Y,, and Ma function of v,, y,; then 
Bored dL dM dM 

Fak i ak be = 0, and mae dL, + pene =; 
and since, by taking the differential equations, we have 
dL dL dM dM 
Sumer dy, =0, and Ta ot: tay od == O, 


by elimination, there results 


dy, 
ne. 
from these equations, and from the equation of limits 
dU,~—3U,=0, which must contain or,, dx,, dy,, dy, 
the values of dx,, dy, must be eliminated, and the result- 


, LY, 
dx,—dy,=0, and eet Ee = 0, 


ing equation will determine the conditions of the problem 
‘that must be fulfilled at the limits. 


The same conclusion will result, if, instead of an 
- elimination, we multiply the equations JL=0, JM=o 
by two indeterminate quantities a, ~, and then add them 
to the equation of limits, which will become 


dU, — dU, + AOL + pdM =o. 


This equation will contain terms affected with dy,, 
dX,, OY, dy,, the coefficients of which are separately te 
be made equal nothing. | 


108 Lagrange’s general Method. 


The origin of the abscissas and ordinates (when the 
discussion is concerning curves) is supposed to remain 
constant; but, it may be supposed to vary, which will 
be equivalent to the hypothesis of making quantities, 
such as a, b, contained in VY, to vary. In that case, if 

iV = Mdxr + Noy+ Pip+&e. + Ada+ BIb+&e. 


we have, as M= hw and N = Line a7 A =>, and 
dx dy 


B= - and consequently the variation 6/V dx will= 


fity.de(w ~ 54 £8 - &e .) 
+fdadzx.d + fdbdx.B 


dQ 
+ &e. 
now the equation to the curve depends on the equation 


ee eed 
N- We Sto &c. and the additional terms daf'A dz, 


dbf B a or da fe ue dx, 5b re ai At will affect the equa- 


tion of limits. 


If c=2' —a, then dx=dr' —da; therefore, if a’ be the 
same, dx = —da; consequently the term 4 da, introduced 
by reason of the variation of the origin of the co-ordinates, 
will be affected with a negative sign. 


re 


The methods described in this Chapter comprehend 
all cases belonging to absolute maxima and minima: 
they extend also, if we employ Euler’s reasoning *, to 
relative maxima and minima; or independently of the 


* See p. 78. 


* * 


Lagrange’s general Method. 109 


last author’s method, which may be thought not perfectly 
satisfactory, they may be extended by the following 
process, which is Lagrange’s. 


Let w be the function, the integral of which is, to 
have, within the assigned limits, a determinate value. 
Let s be its integral, then ds — udx=0; consider this to 
be an equation of condition, such as L=0; then, since 

adL was added [p. 107.] to the variation of JV dx, we 
- must now add 

Ad(ds — udx) .... [a] 

the first of this, Adds=adds=d (ads) —da.ds ; a term 
therefore—da.ds will be introduced under the integral 
‘sign, which, since @s is an arbitrary quantity, must=0; 
~. d\=0, or A=a, aconstant quantity. The term d (ads) 
will solely affect the equation of limits, and since the 
whole variation must be taken between s, and Sy ads, 
_—adés, must be added to the equation of limits, which, by 
the hypothesis of s having a determinate value, must = 0. 


Hence there remains of [a] 1.10, only -aé@ (udz) ; 
consequently, the whole variation is reduced to this 
df(V —au) dx: whence the rule before given, p. 78, 
which reduces relative maxima and minima to absolute, is 
derived ; for the formula directs us to find the conditions 
of the absolute maximum or minimum of /(¥ —au) dz, 
a being a constant quantity. 


We now proceed to the last Chapter of this Work, 

in which, the formule previously established, will, with 

some slight alteration, be applied to the solution of 
_ problems. 


CHAP. ViITk 


Particular Formule deduced from the General one, for the Purpose 
of facilitating the Solution of Problems—Problems solved. 


Evucer in his treatise has deduced from his general 
formula, several subordinate ones, limited indeed, but. 
materially expediting the solution of problems. These 
will be first described. 


In the general formula, 

dV =Mdx+ Ndy+Pdp+Qdq+Rdr+k&e.. . [A] 
suppose M, Q, and all coefficients excepting NM, P, to 
equal 0; then, dV = Ndy+ P dp; but generally [p. 89.] 
N- ee be &c. = O [a]; inthis case, therefore, 


N-S=0; consequently, Ndy—aP =o, or, since 


2) £Y , Ndy—p.dP = 0; and substituting, in 1. 10, 


dV =p.dP+Pdp=d(Pp); whence, by integration, 
Me a ee [b], c being the correction, 


If Mis not=0, then V=fMdx+Pp+ce.... [el]. 


Let M=0, N=0, and all the coefficients after Q; 
then dV = Pdp + Qdq, but the general formula [a] 1.1 1d | 
in this case becomes 

dP  #Q 


—€——_—_ _-_— — O * 
dx d x’ : 


Ce a a ne 


Formule adapted to the Solution of Problems, Ill 
d’?Q dQ 
whence, P = /——d = —— 


multiply this by dp, and since oP =q, we have 


P.dp = ¢.dQ + c.dp 
hence, substituting, in the equation, [4] p. 110. 
dV = 9.dQ + Qdq + c.dp = d(Qq) + c.dp 
and integrating, 
V=Qqt+ecpic...... [d] 


c, c’, being the corrections. 


If M does not=0, we must add, to the above form, 
the term f Mdz. 


If M=0, but MN is not =0, that is, if the form be 


then, since Ne pa =! 0} 
dx dx? 
‘we have, multiplying by dy, which = pdx, 


2 


Ndy — p.dP +p. SS. du = 0. 
. a’Q dQ AQ 


2 


dQ dQ 


—~ 


- 


Hence, substituting, in J. 12. 


dQ 
Mi ndP + Pdp —d(p.2) +9dQ.+ Qdo; 
B= p-d(p.—) +g q 


consequently, V = Pp + Qq - a Gos eeeley. 
If dV =Pdp + Rdr, 


112 Problems. P 
then the formula {a] p. 110. is reduced to 
Me dP id eh i 
dB: ae 
CR 


and Pdp=c.dp— dp. : 


consequently, P=— 
hence, since dp=q.dx, we have, by substitution, 
dV =c.dp+Rdr — q-Gade 
= c.dp +Radr - age HY + dg.5~ i 
[since dg=rdzx| =c.dpt+ Rdr-+rdR—d(q.") 


Hence integrating, and adding the correction ©, 


: AR Oe 
Y= .cp + Br — gia 2 Gee» a Bal 


These forms are sufficient for the solution of the fol- 


lowing problems. 


Pros. 1. 

Required the relation of # and y, such that 
f(ax—y’).ydx shall be a maximum or minimum. 
Comparing this expression with (Vda, V=axy—y*s 
consequently, dV =ay.du+(ar—3y’) dys I 
which, compared with the general formula [4] p. 110.” 
that is, with d’=Mdx+Ndy+Pdp+Qdq+&e. 
gives M = ay, N=ax—3y’, P=0, Q=0; 


dP 
consequently, since NV — pr &e. = 0, in this cas 
y 4 
ax — 3y° = 0, andy = 3° 


Problems. 113 


Pros. 2. 


Required the shortest curve that can be drawn be- 
tween two points, or between two curves. 
Here fds or fdx/(1+p*)=a minimum; .«. V= 


/(1+p") and dV = waiean ; WM, N, &c. =O: and 


= ; consequently, the formula [a] p. 110, 


a ae 
VJ (1 +p’) 
© Sab y pct lls aaa aaa eee i 
somes d( ap) =° p=a/(l +p’) 


a 


and or — mB whence, by integration 
P ae Ved 6 gy (1— a) 3 3 y S 2 


y/(i- ge =ax-+c, an equation to a right line. 


Pros. 3. 


Required the curve of quickest descent between two 


given points. 


zi. time= fF ae ash, fle) ie) 9, 


VSP which compared with fVdx gives 
= Vier). whence dV = — vi LUTE) dy + 
Vy 2y 


Ba ae) ‘p, which compared with Form | 4] p. 110, 


Vy 7y7 OEP) 
a Aas 
igives M=0, N= — oy? Gale J/ (3 me +p yy? 


Now by the Form [6] p.110, Y= Pp-+c; .. in this case 
a 


114 Problems. Z 
A HE 8 le Yd 
vy Sy OTP) 
whence J Matai = cC, or ee = 
VIS +p JS yVv (dx*+dy’) 
cr/y 
and finally, dx = ——— S/T = ey) .dy, an equation to a cy- 


cloid. See p. 5. 


Prop. 4. 


Required the relation of « and y when f(a’ + y7)”.ds = 
amaximum [see Euler’s Methodus Inveniendi, &c. p. 52. | 


Here V=(a" +y*)" x / (1+p"),sinceds= da / (1+p*) — 
and, dV = 
(+ Y)"P ay. 
J+) 7? 
*, M=2nex (2’+y)"",/(1+p"), N = 2ny (a? + y?)"" I 
| (a? -+y")” p 
Lise pr) and Ee eye 
x/{ P ) ath yee +p’) 


Here it is more convenient to employ the general — 


(Qnxdx ae ony dy)(x’ +y’)" "7 (1+ p’) + 


formula, 
N _ dP + &e. = O, 
dx 
which, since 
APE TYT (aia ae A yg 
Facsp) (eee npydy +e TES ge 
by reduction becomes 
an(ydx—ady) dp __ rh] 
wma Lp : 
if §=are, whose tangent = = dhen dt, dx— ady | 


> Problems, i153 
consequently, integrating the equation [h], p. 114, 


; | d 
2n§=A+c, A being an arc, whose tangent=p, or 7 


= tan. #=tan, 5 — ~ (A +c). 


Pros. 5. 


Required the curve which by a revolution round its 
axis generates the solid of least resistance. [see Newton, 
p. 324, 3° ed. Euler’s Methodus Inveniendi, &c. p. 51. 
Simpson’s Flutions, p.487, Ed. 1750. Emerson’s 

_ Fluxions, p. 183, 3* ed. Lacroix’s, Calcul. Diff. tom. II. 


p. 698. | 


The resistance = =f 2. fae , which, compared with 


[V dx, gives 


Lp 
P Syp typ ne 
pati y pet Ory Pi 


3 p* + mt 
consequently, M=0, N=. eee P Sacre : 


Hence by the Form [6] p. 110, using —c for the correct’, 
BPH MORBY PIER Y PE 20 5 
Ly aC -+Z ). 

whence, by reduction, 


oh. 
¢.(1 + p?)* = 2yp®, and y = aha UBT 


116 : Problems. , ‘ 
3C c c ) 
but, pdx = dy = — -odp —— dp + 54p3 
3¢ c c dp 
hae ie Ln pry. > 


5° 5 .log.p+e 
8p" 2p? VP ge Eer 
which equation must be laa with c(1+p*)?=2yp’. 
Pros. 6. 


Required the curve in which fyxds isa maximum, 
[see Euler, Methodus Inveniendi, &c. p. 52. | 


Since, yxds = yx, /(1 + p’).dxa, V = yx,/(1+p’) 
and dV =y,/(1 + p*)dx + a ,/(1 + p*) dy + | 


yp 
(1 +p)’ an 
EAL ox a) oN Pe 
yf (1 +P"); ba ee, 


/ +P) 
In this case therefore, we must use the Form [c] p..110, © 


and then yx, /(1+p*)=fy,/(1 tr ldet 7a + 


Wekurer ee 2 
=c+ fi 1 + p*).dzx; 
which by reduction leads to a differential equation of 
the second order. 
Pros. 7. 


Required the curve, or in other terms, the relation of- 
ef E 
x and y when vf =a maximum. (See Euler, Comm. 


Acad. Petrop. tom. VIII. p.171.) 
dy d’y _ay dx Np gee 
DNASE a) 


—_—— 


ds dxJ(i+p) dx J +p’)? * 


" Problems. 117 


ddV= = —2_ dn + + 


. M—o, N -0, P= — GP ¥ Ga ce 
(1 +p*)* J (+ p*) 


.. by the Form [ d = E14. 


=~ en tc, 
JU4F +p’) = Jaw Wig aloes 
whence, p or “ = — = and cy = c” — c’x: an equa- 


tion to a right line. 
Props. 8. 


dl’ : 
Required the curve in which ye 7s =a maximum. 


[see Comm. Acad. Petrop. tom. VIII. p. 171.] 
ay d*y 


Meee eo A) Vie ae 1) 
y.ds dx.y/ +p) yV (1 +p’) 
LEARNER OY DEIR 7 ean 5 gy 278 

NS g/tp) y(l +p)? ” 
1 
eee ——— 1 77 « 
Beit). 2 
qP 
ee M=o0, N=- A | Sp bed ne ae car 
| yVv (+p) y.(1 +p)? 

1 
ee. oh itiy the Form. [el] pe tlt; 
YJ (+P) ‘ F 

y pap 

hel abe Me, oe 
[ since = yf (1 +P’) wer 

yJ/(+P*) ya +p')® Jae) 


AEST ee 
YJ +P) — y(h +p") 


tl arta neal 


118 Problems. 


consequently, =-es7and 


det dls sce 
VERNAL + PG 


.: y f2.dy 
and since p = as rf yJeypte(4+ey) +0, ](440y) 


Pros. 9. 


Required the curve which, within its own arc, its” 
evolute, and radius of curvature, shall contain the least. 
area*. [see Euler, Comm. Acad. Petrop. tom. VIII. 
p. 169; also Methodus Inveniendi, &c. p. 64. | 


a ds3 _ (dx*+dy’)* 
The radius of curvature = ate Pi “Eee 
a peas wpe te GAP Gar ey A eee 


radius x ds 


Hence, the differential of the area = % 


“ On 7 +p*)di= ~ LAP de, which com- | 
q 
pared with Vda, gives V= tr, consequently, 3 


Ay page 2 (1+p')-dp + a iy! 


ee ee ee ee 


* The area is AOR in Simpson’s Diagram, p.78; Fluxions, 
ed. 1750. 

+ Simpson, p.72. Vince, p. 149, First Edit. Woodhouse Anal. 
Calc. p. 180. 


| y Problems. 119 
fi M = 0, N= 0, P —_ “P (+P*) Q- (1 G5 eae 
q 
*. by Form [d] p. 111, that is, de +cp +c’, we have 
(1-++p?)? (1 +p?) 5 
Sees ee ee ote Cy 
g q P ; 
2(1+p*)’ d 
or — EP = pte; but g =F, 


. — 2(14+ p*?)FV (1 + p’).dx = cpdp + c.dp; 
or, since, ds = ./. (1+ p*).dx 
a3 as ot Ti a + ce 

Gihgtyaieasl (hap )* 


3 


C c.p 
*.—-25=—- — + Tp SRT ~f. 
af Pai WY Op) 
2 cp —c y, 
Vit+P) 
If c,.c’.= 0, —2s = PEE, which shews the 
J +p 


‘| CY. ite pea! 
curve to be a cycloid: for by p. 5. pe ce Vv ( a 


. Gasp = ty {Ser 12 and the are of a cycloid 


measured from its vertex = 2 chord of generating: circle 


= 2,/a (a—y) = erat 


Pros. 10. * 


2 7 ' wey e: ie: 
Required the curve in which ay ae a isa Minimum. 


[see Euler, Comm. Acad. Petrop. tom. VIII. p. 185. ] 


— ee 


J20 Problems. 
Ls ees eu 1 ee 
Here dx.dy = de x pen & 3h =p 
, r ] é 
and dV = ——.dp +-dr; 4 
iL % : 
Ma0,.N=00P=\~ 0 sien ee 
P P 
and by the Form [ f] p. 112, 
- = cp +--+ SSD 
ey P p tdaxp fs 


Hence, 0 = cp +f — c, and g=p,/(c'—cp) 


dp dp 
or, =- = @-.cp) and .. af = — =e 


nd isd geidis RY A eM Airey? 
RU Chie A Wy CERI. CES hep) 


Pros. 11. 


: oe Ly * <P \eoy So 
Required the relation of w and y when S- da a 
maximum, [see Lacroix, Diff: Calc. p. 704: also Borda 


Mem. Acad. des Sciences, 1767. p.560. | 
(CW = (F) dea Gpdasy VV = 9, 


dx} 
and dV = 2q.dq; ~. M=0, N=0, ‘P=0, Q=249, 
R= 0, &e. 


| 
| 


PQ _ | d’ 
‘. by the Fonte [a] p. 110, ae ; au nt 
and eee 2.44 = = 0 uaeep Or och = C2 (0 


2p or gl oe + cx +c’, and 
i dx 2 P 


i* Problems, 121 


, 


cx Cx 
2y = cx ne 
eo ag Tee 


The final equation contains four arbitrary quantities, 


¢, c’, &c.: in order to determine them, suppose the values 


of y to be given at two points of the abscissa (x=0, 
_#=a), and also the angles in which the tangents to the 
curve at those points * are inclined to the axis; for the 
sake of simplicity, suppose y=0, when #=0; and y=6, 
when «=a: also the values of the two tangents of the 
angles in which the curve cuts the axis to be ¢, #: then 
we have, for the determination of c, c’, &c. the following 


equations: 
€=90, Yy=0; Shc ats 0; 
P= aay = bh) bao oO 4 ca, 
05 Hat; ro Re Se; 
eaa;e. Hae; ot =D car ¢, 


whence the values of c, c. 


The preceding problems involve one property only ; 
that of the maximum or minimum; and therefore, in 
strictness, ought not to be classed amongst Isoperimetri- 
cal problems, since they involve neither the Isoperi- 
metrical property, properly so called, nor any other 
equally affecting the theory and the analytical pro- 
cesses. The following problems inyolye more than one 


property. 


« This is only one of the many hypotheses that may be framed for 
the determination of the arbitrary quantities, 
| R 


bd 


192 Problems. 


Pros. 12. 


Required the brachystochrone, that is, the curve of 
quickest descent, when the length of the curve is given. 
[See John Bernoulli's Works ; tom. II. p. 255. Acad. des 
Sciences, 1718. p. 120; also of this Work, pp. 18, &c.] — 


Here, by the formule, pp. 78, 109, we must find the 
variation of fdx (V — au), puttmg VY — aw instead 
Of ik: now 4 Fes a and w= ,/(1 +p’); 


chee Ua Aen aw ty 0p | ~— a,/(1+p*); 


eT Ti oe ny og t iM 
La pees +p Fed Cr * a) rap Pe 
P 
= swe), P= CH -0)* Fa 
;. by, thesForm, [5] p:.£10. 
OW fal eile eee J , 
WAC! 1?) 7,.8).= oa (air) + C3 
or e — a) = ¢,/(1 +p’), and by reduction, 


Tie cr/y dy . 
J Ll 2a/y + (a — ey] 


If, instead of the length, the area had been giver, — 
then fudx = fydx; and consequently, we should have 


WV’, or V-au = S+P) 


Siena 


in = a case, P = Si aE Sh 
SYS (+P) 


Problems, 123 


VAC aes ee ee 
and, v9 Y= Ty JU +P) a Oy 


whence, | ery =c-+ ay, and by reduction, 


: (c+ay)/y.dy 
J Li-y(e+ay)'} 
ay? _ayrdy — 

J - @y) 


son’s result, Fluxions, p, 492. But, since c is not neces- 


If c=0, dx = which agrees with Simp- 


sarily=0, it follows, that Simpson’s solution is not so 
general as it ought to be. 


Pros. 13. 


Required the solid of least resistance amongst all the 
solids of equal capacity. [See Maclaurin’s Fluxions, 
p- 751. Emerson’s Fluxions, p. 188. | 


Here, [see Prob. 5. p. 155.], Va 2 
= fry dz ; 
*; V—au, or V’ =f. 


p 
ern and fu dz 


— ay {including # in the 


iP 
quantity a.) 
‘Hence, P = SEP , and by the Form [5] p. 110, 
3 
oP sash sah dae c, whence, by reduction, 


ate” CU py 
(c—ay*) (1 +p")? =2yp’*, or (c—ay’) ds* = 2y dy’. dx. 

If instead of the condition of equal capacity, that of 
equal superficies be substituted, we have 


194, Problems. 


z 
eee 


V—au, or fan — ay, /( + p’)s | 


anit ued aes Sup + yp" ayp 
G+PyP” Veit. 
whence, by the Form [6] p. 110, 
YP » —_ SYP +YyP” ay p 
naw Wistar Viered 1 + = — — or 
tp UN I e/a 
and, by reduction, 
c.(1 + p*)* = 2yp+ay.(l +p 3 
or c.dst = 2y.dy>.dx + ay.ds*.dx. 


— Cy 


If the solid of least resistance be required with both 
the specified conditions, then we must use a formula 
f F —au— by) dx. See p.78, and if w=y*, and v= | 
yf (e+ 1+p*), we have | 


P-ou—os, Orr = —~ ay — by, /(1 +p"), 


1 es yee 
and consequently, by the Form [6] p.110. 
fg BYP YP byp* 
eg aren Ja+P) ~ c, whence | 


c(lt+p')? = 2yp'+by(1 +p*)* + ay*(l + ys | 
or (c — ay’)ds' = 2ydy'dx + byds*.dz. 


Pros. 14. 


| 

4 

Given the length of the curve, to determine its nature: 
when the solid generated by its rotation isa maximum, — 
[See Simpson, p. 486: Maclaurin’s Fluxions, p. 7495 | 
Euler, Methodus Inveniendi, &c. p. 196. | 
[Vd«x =fry.dx [r= 3.14159 &c.j and fudxr = 
f/Utp de; 


, 


Vv’, or V — au = xy? ~ a,/(1 +p"), 


3 
Problems. 125 
ap.dp 
dorrihetieten, Wein Codes m19 = —— 


(1-+p*)’ e+ Pp)" 
*, by the Form [b] p. 110. 


-anddV’=2ry.d 
0” TOP 


j F ap* 
-af(l+pr)=c- —~—-—, 
wy (1 + p’) JOtP 
Shag +P.) = , and by reduction, 
dy _/{a'— (ay 8) (ry’ —c) 


ats Menten [a= ey or] 


ry 
V (a — 


result, and, like the FU Spin restricted. 


i G.s2 0, dies yi) > which is Simpson’s 


Euler says, that the curve is the elastic; and the 
curve determined by Simpson is the rectangular elastic 
curve. 


Pros. 15. 


Required the curve that generates the solid of the 
least surface, the area being given. [See Euler, Methodus 
— Inveniendi, &c. p. 198. ] 
V=Qry/(l +p’), v=y; 
Vv’, or V — au=2ry//(1 + p’) — ay, 


Sand dV" = [29 J +p") a] dy + FP ap, 
; whence P = Pa <A ae cies by the Form [5] p. 110, 


J (i+P’) 


/ Qry 
oe) eg ae +e 


=e 


and, by reduction, 


nt OR: 


126 Problems. 
a / [4ey* = fetay)'] « 
c+ ay 


c+ ay 
WAR A ae "(oneal 


p, or 
or dx = 


If instead of the area, the length is given, then | 
fude =fds = fdu/(1 +p’); and w= AL +p?) 
 Visary /(l toa LAAd +p*), and consequently 


= ero] 7) P and by the Form [b] p. 110, 
(27y — a)p* 
ory — 1 ty) Se ee 
(2ry—a),/(1 +p’) J (i +p*) te 


whence, by reduction, 


Pree cdy 


Jf Lary —a)*—e*], 
and w 7 hyp. log. [(27y —a)+,/(2ry —a)?—C] +c, 


and the curve is the catenary. [See Euler, Methodus 
Inveniendi, p. 198. | 


Pros. 16. 


Given the length ofa curve; required its nature, when 
its area is a maximum. [See p. 41.]| 


Here V’= //(1+p’?)—ay; -. P= Wier and by 
the Form [6b] p.110, 

es ehh pinata 12 Spice 

MOP ay = Toppy ts aap 


and p, or oY oe, )'] hence, 


dy, and x=c' = ./[1—(c--ay)*] 


=c+ay, 


oy ted iis. anaes 
J Li-(e+ray)*] 


an peel to a circle. 


Problems. 197 


Pros. 17. 


If from a point two straight lines be drawn, inflected 
to each other, and their extremities be joined by a curve; 
it is required to find its nature when its length is a max- 
imum ; the area included within the two lines and the 
curve being given. [See Euler, Methodus Inveniendi, &c. 


p. 136. | 


Let a line drawn from the fixed point to the curve, 
(usually called a radius vector) = y, and let d, 
(rad. =1,) be the elementary angle between two contigu- 
ous lines, y and y+dy, then the arc= /\/ (dy +y’.d@). 


oo = , dx being 


- But the differential of the area = 
dx 


invariable; .*. dé? = as and substituting 


SV (ay +S) = a maximum, or putting p = oe 
SV (v' +2) 2 =. a maximum. Hence, V = 
SOOM iBT Eat 
V(r u p an GPA): 
-and by the Form [6] p. 110, 


V (9 +a) Jo ab to 1=cy,/(y*p* +1); 


dy /(1-c’y’) dx c.dy 
Hence p, or on oy y. Vue) 


the property of a circular are described on a diameter = 


' 198 Problems, 


b ; the fixed point, from which lines such as y are drawn, 
c 
being in one extremity of the diameter. 


Pros. 18. 


Required the curve that, by a revolution round its 
axis, generates the greatest solid under a given surface. 
[See Euler, Methodus Inveniendi, p. 194. | 

Here V’’, or V — au = ry*® — 2ary,/(1 + p’); 

; Apts 2aryp 
a a J +p’ and by the Form [b] p. 110, | 
: 4 Q2ary p” 

79 ! eh Om 
ry? —2ary,/(1 +p’) a uiBhatae | 
putting cw for the correction: hence, by reduction, 

_ Vitey—(y'- 9") | 
a or Ky y*—C 
$y rem 
~ / 4a — Y= 0) 


the equation to the curve. 


NE Rrarmat s Pua oP oe oe an wequaten to a circle : 
fs / (4a?—y’) ays 
which Simpson [ Fluxions, p. ae determines it to be; _ 
but, itis plain that this is only a particular case of the 
general solution. Lyons’s Solution, p. 100, is also equally 
restricted with Simpson’s. 


Pros. 19. 


Of all Isochronous curves to find 4B such, that the — 
space included between the are 4B, and a chord drawn 
from 4 to B shall be a maximum. [See Jobn Bernoulli's | 
Works, tom. II. p. 263; also Acad. Roy. des Sciences, 
1718, p.132.] 


Problems. 129 
Here the time, a given quantity represented by fuda, 


answers to fd a > 


and {Vdzx, the maximum, to 
J fyte aC). ow Meee, 


or Sz — S) dx, since p = ay 


da’ | 
Hence, V — au, or V’= : (y — xp) — a tp 3 
2 WY 
Sih 2 N= sae) nd Pe 
a EN RS a 
Rein FP oe 
.. by the Form [c] p. 110, 


OC (Ah) SE AGRE 2 Uy F 
3 Y-tp)—a Sy Thre Vy SL +P) 


dy | . fpdx = y. Hence, by reduction, 


But eres 


a 


130 Problems. 
dy _J(@-yly-¢)] 


a 
dig ad «anu adh apr ee 
DO SGA EER) Redan AO) 


and consequently, 
ees 
V/[e—y-(y—c)’] 
the equation to the curve. 
Pros. 20. 
Of all Isoperimetrical curves drawn between B and 


N, to find BEN such, that BZN shall contain the great- 


est area, PZ being a function of PF. [See James and | 
John Bernoulli’s Works, p. 909, and p. 816; also pp. 12, 
25, of this Work. | 


Let PF'=y, and let PZ= a function of y=¥; then — 
JV dx corresponds to fF dz, and fudztof/ (1+p*).dx. 


Hence, V + au, or V’ =F + a,/(1 + p*) 


A. d¥ a : a 
and dV’ = —— (2 -|- Bk: cee, 3 oe P SS Pas iiss, 
i LE nN J (+p) 


Hence by the Form [6] p. 110, 
5 ees OT 1 + gh We PET) 
’ / ( Sw Ae vY (147) sn C; 


ua 


Fat = ¢ — ¥, and by reduction, 


or, 


Problems. 131 


| ee a 
p> or = la serge 20 y'] , and consequently, 


CcC— 


dx= rACHtan the equation to the 


curve. 


In the equation, as it stands, there are two arbitrary 
undetermined quantities @ and c. The integration of 
the equation would introduce a third: and to determine 
these three, we have given, the two points B and JN, and 
the length of the curve. 


Pros. 21. 

Required the curve which generates by its rotation 
the solid of the greatest volume; the length of the curve 
and its area being given. [See Lacroix, Calc. Diff: et 
Int. vol. I. p. 713. | 


Here we must use the form [See p. 78. | 
ve (V — au — bv).dx, corresponding to which is 
Sly? — ay—b../(1 + p*)] ae, 
a and b involving, in their values, + and other given 
quantities. 


Hence, the differential of the quantity corresponding 
to V is 
dp 
2y — a).dy — bp. ———; 
(2y — a).dy — bp Jatp) 
“. by the Form [6] p. 110, 


2 


bp 
ee ey; 1 >) — p ae 
y ay aya + p’) C nt EP) 


Yay -C= JaEP , and by reduction, 


132 Problems. 


biel eee 3 
yo —ay—c 
won (yh ay—0) dy 

a= 7 (PY aay-F 


an equation to the elastic curve. 


p, or oY and 


Pros. 22. 


Let now PZ, instead of being a function of y, (see 
Fig. p. 130.) be a function of the are BF (s). [See 
James and John Bernoulli’s Works, p. 912, and p.917: 
Maclaurin’s Fluxions, p. 508, also p. 25. of this Work. ] 


Here f V'dx corresponds to f Zdx, Z a function of s, 
and fudx corresponds to fdx,/(1 +p*); 
. V — au, or V’= — a,/(1 + p*) + Z, 
RACH eee ie, See ae 


This form therefore is to be solved by the method 
given in p.93: and, on comparison, we have p#=0, 
ym bes Ps AEE = 
v=0, 7 = ~ FaER Tp)’ T= —, ds=dt. The formula 


of solution then, p. 93, is 


d 
ad ) a sae abo 2°6 
k beng = H ~fT'dx. Hence, integrating 
pd hd yal So wi 2 


ACES 2 Toke yf ee oN dc 
p a Nee ane 


&. LdeY (1+p*), or Tds = — cc. a: . fT ds, that is, 


Problems. 133 


mi? ay eC 
fadZ or Z=C + = Hence p or a. gw and 


Ifdx, dy are to be expressed in terms of the arcs, 


then since da= dy E=9) ,ds= Vr ~ (F2 Sig dy 
c.ds 
VY [e+(Z-c)’| 
(Z—c’).ds 
5 ger er 
"= / [e+ (Z=e)"] 


consequently, dy = 


Pros. 23. 


Given the length of the curve; required its nature 
when its center of gravity is most remote from the axis 
(y). [See Euler, Comm. Acad. Petrop. tom. VI. p. 146: 
Simpson’s Fluxions, p. 498. 


Suppose y to be horizontal, and a vertical, then the 


distance of the center of gravity from y, = of a *, 


But fds = s = b by hypothesis; 
er Te VAO i 
Bake au, or V/V! = are —a,/(l+p’), 


and P = G -- a) x aa 


Hence, by the general formula, WV — _ + &e. N, Q, 


* Simpson’s Fluaions, p. 205. ed. 1750." 


134 | Problems. 
&e. being = 0; we have 
os fo: Fe 
alG a) J (1 on : 


pO OM eee ed rs — dp 
a ee Se ee , and dx = be *p/SL+P) | 


Hence, dxf (1+p’), ords = — oft si ? andintegrating, — 


CED Oe 

sabe, or p= 3 sonora 
(s—c)ds 

/ L(s—e)? +0°b" |’ 
ch.ds 

and Y= Tle ey eee]? 


equations to the catenary. 


dx = 


If the condition of ia Ane be omitted, that is, 


if a=o0, P will equal = ; and the nature of 


b JSa+P’) a +p)’ 
the resulting curve will still remain the same, s being 
represented by 6: and if s be not so represented, 
but be supposed to vary, still the resulting curve. will be | 
the catenary, as will appear, with the reason thereof, in 
the following problem. 


Pros. 24. 


Suppose a chain or cord, of variable thickness, to be — 
attached to two fixed points ; it is required to find the 
nature of the curve which it ought to form, so that its 
center of gravity shall be the lowest. 


Let the weight of an element of the curve be ex- 
pressed by @.ds, 0 being a function of s dependent on the 


Problems. 185 
law of the chain’s thickness; then, the distance of the 
fads 5 fuxb.ds 

J ‘ds ” fo. ds 


center of gravity, instead of being swhich 


is to be a maximum. 


fxbds _f9.ds.d df xdds— fxb.ds.dfb.ds 
Hence, 3 Fide) = baa TM LEAS a ule = 0, 
now dfxd. ds=tfx)/(1 +p"). dx; ; SA eee LV 0 -+-p’) 
and dV = 


. Qxp.dp » ab 
O/(l+p ) ML Tt ee) rhs t / (1 + p*) 7, ds. 


Hence by the Form [C] p. 93, supposing dv =0, 
and not regarding S, the sum of the definite terms, 


o(Vde = fdv.dy[ — “(zie | | 


Sia le fs (item igh di a) 
giice a Tat)’ N= =F VALET OL an 


k, which equals H—f{Tdz, being in this case = H — 
» a ‘pe 7 
Sav (L+p’).7.da =H-fedi=H— xi +fi.dax 


substituting this value in the expression [m]; the 
equation becomes 


d 
dfVdu = fdx.dy. [ - (Fee q Bb ed ey =| 


With regard to the other variation 3/6.ds, V answers 


Opdp 2 19 7. 
B00. (itp) and dh = 7a VEEP) geass 


136 Problems. 


* by Form [C] p. 93, supposing dv=0, and not 
regarding S the sum. of the definite parts, 0f/Vdx = 


yes: re kp 1 
fae.dy| —d Tray hp — d. / (+p) -- | > % [7 | 


4 9p 
aig 8 


equal to K — f/f (1 +p). <des that is, K — fdé, 


or K—6; K is analogous to H, and equals the entire 
value of 6 between the two limiting values of x. Hence, 


substituting in [v] 1.3, we have 
= o2 
dfi.ds = fdz. Ae d( sates) a 


If we now substitute in the expression for 


B) (! ra) p. 135. 1. 4, and represent the two integrals 


of w0ds, and éds, taken between the two limiting values 


fx, and x,| of 2, by A and B, there results 


Al dx 
Bfdsty | - d(— a ears i saa ae 


ee le 1 TaF ey): 7.| =0 = 


and putting 3 =A, 


Hp+p fide K 
a (BABIES) n.d (RP), and integrating 


Hp + pfid« = aKp + c,/(1 + p*), whence 


and #& in this case being © 


Problems. 137 
H-aK+ fide =c VOTE) and taking the differential 


¢ AP 
P/(l +p)’ 


d . 
a DE ONT 6 rl ; whence integrating 
4 


dp 


Odx = — or 6dr (1 +p) = — Pr 


Cc 


dy 
fds = ate, Bd Ye er aaaere 


dy? Ss 
Ba since yitr=V(1 + 73) = : 
a fads —c' 
we have dx = St fussopaey 
cds 


and dy = J feds—cy +e] ’ 


If the chain be uniformly thick, $= 1 and 


ae (s—c)ds 
~ AS U(s—eP +e)’ 
ee cds 


which equations are the same, in fact, as those in the 
preceding problem, [p.134. 1.67], and belong to the 
‘curve called the catenary. 


In problem the twenty-third, where s is a given quan- 
tity, the solution was effected by means of the common 


dP 


formula N— fae + &e.; and the same formula is suf- 


ficient, if instead of the process there used, we institute 
one similar to that in Prob. 24, p. 134, that is, if we 


; 
= 


138 Problems. 


feds 
fds 
sx dfads—fxrdsdfids 


Ss 


investigate the variation of 


—afds, which equals 
— adfds 


or a[s8/Tey/( + p?) |dx - (frds+as*)df, (1 + p*)dx | | 


for, it is manifest, that this expression can be expanded 
by the common formula, p. 88 *. 


The condition however of a variable thickness intro- 
duces the quantity 6, a function of s ; for instance, instead 
of dfx./(1 +p*).dx—&e., we have, [see p.135. 1. 5. ] 
dfx i, (1 +p*) — &c.; and therefore when the differential 
of the quantity corresponding to V’ is taken, a term 


x /(1 + p*) Gods is introduced, which obliges us to 


—— Se eee ee = 


have recourse to the formula [Cc] p. 93. 


The next instance will illustrate the use of Lagrange’s 
formula in his third problem, p. 185, in which, the quan- 
tity that is to be a maximum is expressed under the 
form of a differential equation. 


Pros. 25. 


Required the curve,.down which a body falling, in a 
resisting medium, shall acquire the greatest velocity. 


* Euler, Comm. Acad. tom. VJ. shews, that the catenary equally — 
results, whether the length of the curve be taken into the computation — 
or not. ‘* Potuisset quidem eadem equatio multo facilius inyeniri, si 


leds 


neglexissem denominatorem, quippe per priorem conditionem | 


debet in omnibus curvyis esse idem : Verum quia hoc fortuito accidit, &c. 


Problems. 139 , 


Let x represent the square of the velocity, g the force 
of gravity, e a function of z, the resistance ; and let x be 
supposed vertical : then, 

dz —2g.dx + 2e\//(1+p’).dx=0,... [1] 
[by the principles of mechanics. | 
Compare this with the formula p. 94. 1. 12, and we have 
V=2z,X=2g, U=2¢\/(1 + p’) a function of e and p; 
dU dU 2ep 


Pa iicicn a Ag tical eh loge aii 
dp ie de © ~ /(i+p*) A 


d 
aot dz /(1 +p’) 


: — 2ep re Fie doe 1 ya\ ae 
Pad O\= Taps 8 and P=27/ (1 +p”): 
hence, the equations are, 
da d 3 
Se ae yep) = 0.4 7. 2 


BGs Or Ata Ogee 7? (eT 7,30 = 0, 


/ p 1 
or 2Ae =O Ct et wt ee 3 
In equation [2] substitute from [1 | the value of dz; 


.; 2gdn = 2af/(1 +4. p’).de oo 2e/f/ (1 + p*).dr 
=2f/(1 +p’) [ade+e.da| =D (E +p?) d(ag) = 


d 
2.d[ae/(1 + p*)] - 20 aa = (by [3]) 
2.direv (ep) ~ Fe 


Hence, integrating, 2gA= 20 V (1 +p*) — ez +c= 


I+? p 1 
—— —- —>+t+enar — +06; 
P»/ a Jf a a/ &+pP 


and .*. from (3), 5 e A be 


e ye ae 
SOUFEP) E/E TP) fa 


140 Problems. 
whence, by the solution of a quadratic the ‘Value 


of p. 


This problem may be also solved, and readily, by 
means of the general method and formula given in p. 104, 
thus: since x is to be a maximum, { V’'dx corresponds to 


fF de; 


why ee “(F .*. by comparison, x = 1, all the other 
coefficients Pane =O sits, Gh Sy. Qua we eae A 0, 
Z = 1. Again, 


JS W dx corresponds wf [ — 2g + 2¢.,/(1 +p’) | de 


.. by comparison, 


ep ay 2G 
= Jaepy 72 +P) Be w= 


E Py oad fa) d(a P’) Los 0 2Aep q 
: up on ir ae aC FG Pt da? | 
VAs ih a ‘ 
CAL Saad iv jal Dee 
(Z’) = An =A. 


Hence, taking from the expanded form, for 
df{Vdxt+ofraWdz, or | | 


dz dz 
Of zd + the [qa 2EA4 207 (14 7") | dx, that 


= 


Problems. 141 


part which is under the sign /, and on which the nature 
of the curve depends, we have 
(Y').dw + (Z)dw = 0, and (Y')=0, (Z)=0, 
since dw, dw’ are independent 5 each other ; ae: 
2QrAep 
7 é ee ee 
(Faery) =0, and 205 E/ (+p) - == 


ea before, p. 139. 1, 12, 14. 


The following problem will illustrate the use of the 
general method given in p. 104, in which the maximum 
or minimum property involves a quantity not expressed 
except by means of a differential equation ; which equa- 


‘tion, in the subjoined case, is called an equation of 


condition. 


Pros. 26. 


Required the brachystochrone, or the curve of quick- 
est descent, in a medium resisting as any function of 
the velocity. : 


Let z represent the square of the velocity, g the force 
of gravity, e a function of z, the resistance ; and let y be 
horizontal and 2 vertical: then by the principles of 
mechanics, 

dz — 2g.dx + 2¢.ds = 0, 


dz A 
or [2g +2e/ (+P) | dx = 0 


the differential equation of condition. 


If we compare this with Wdx, p.104, we have 


SOAR ah on ann (Ode sae 
dW = 2.7-,/(1 +p’) dz + (7) 18 ‘Ja +P)’ 


149 Problems. 


d 2 | a day (qs 
a man +p"), r= 1, since d(=) =p 


CSR pwn ; 

SL ahiio Gas POPs : 

a) a 0 ae ; 

and (Z) =av — =2a,/( +p a dix” t 
| 

The equation of minimum corresponding to, [Vd is 
SEP gps Be 
dp BAK LOPS) . 

Sars Pe OP), 
rf Bf (1 +p") 2 Bt | 

and by ah a 
1+p° | 

N=0, P = =0, ke, = — EP) 
UTEP +p)’ 2 

dP p I : 

and F = N—- —+&c. = no 
is, setter DO tes na : 

dx xe 


Hence, substituting in the expression for )fV dx + | 
If(AW) dx p. 104, and retaining only the parts — 
under the integral sign, which must separately equal 0, 
we have ; 


_2rAep VL 


AW Tere +p" )- iicers +p’) a 


vote +) da 
Z+(Z)= Dor rary (itp)st-P=0, 


yes (¥) = 0,or — 


\ 
EE S 


Problems. 143 


1 F 
If we put ¢ = fa + 2A¢, and integrate the first of 


these two equations, it becomes 


. dr de 1 : 
the second is 5 Ps an sa) Ot?) = O°.. [2] 


Add to these dz-—2g¢.dxu-+ 2e.dx,/(1+p’) =0.. [3] 
and the equation to the curve is to be determined from 
these three. 


dt d 
From the value of ¢, — = ———+ 2—~.p+2aA—; 
ceva 9 eh dz § 4 dz 


‘ : ] 
and from this equation substitute the value of ~——~ + 
7 Pe 


ar<e, into [2], and there results 
dn dan dt ; 
ar (2¢7--=-) Ji +p’) = 0. 


In this, substitute the value of dz derived from [3}, 
and there results (two quantities destroying each other) 


dx dt f 
es va Tw +p)=0.... [a] 


‘dt “9 1 a 2 pap 


2° d 
=d[t/(1 +p*)] — ne 
(from the value of ¢ in equation [1]; ) 


hence, the equation [a] is 


: » aA [t./ (1 +p") ] dp 
22 oer d oe aie ; and integrating 


_— 


144 Problems, 


2grn—t,/(1 Ree, = C; 
or, substituting the value of ¢ from [1] p. 143, 


IgA —~ 


nee. id e 1 ~pAZ UL +p’) 

Lie RM Ao Sg Coat OF pa . ) 
from equation [1], and consequently, 
Dy) he") Le tC +e); 
Je Peg Np/e 7. 


from this equation obtain Fie value of dz, which will 


involve the differentials dp, and 7 de wd z. By means of | 
equation [3], eliminate dz, ig ce will result a dif- i 
ferential equation involving dx and dp, or SY .dx: which 


will be the equation of solution. 


In the case of a void, or non-resisting medium e=0; 


hi LON bees ah 2 se P = eee 
yee BM TRE Ri 
ete: a equation to the cycloid. 


e ds/z ait i 
LSee pp. 5, 21.] 


The preceding instances are sufficient, it is hoped, | 
fully to illustrate the nature and use of those formule of — 
solution invented by Euler, on which the nature of the — 
curve, or the relation of x to y, depends. But hitherto, — 
no illustration has been given of those other formule, 


Problems. 145 
that, on assigning the initial and final variations, have 
definite and absolute values *. These formule, it has been 
already said, were invented by Lagrange, and their use 
will be shewn in the following cases, which may be 
viewed, as unresolved parts, or undetermined conditions 
belonging to certain of the preceding problems. 


Pros. 27. 


If the brachystochrone Ff to lie between a point and 


a curve, required the angle in which the cycloid must 
cut the curve. 


By Form [Aa] p. 89, and since the quantity under 
the integral sign f} = 0, 
Vox, + P,.d0, = 0; 
for, the initial point is given; .. dx,, dy, = 0, and dw,= 


dy, — polo. = 9, and since 


pV), Q=0, and P= 


[see p.113]. : 

Hence, (VY, — P,p,).dx, + P,.dy, = 0, 
bbut V= Pp +c [p.110]; . c.82,+ P,.dy,=0.. fm). 
Let now M (a function of x, and y,)=0; then, [see p. 107. ] 


dM dM , dy, 
qe 4 AS Rami, and eos a JY, = 0. 


p 
hilen/ Ot aye 


Eliminate dv,, dy, by means of this and of the preceding 
equation [m], and there results 


* P. 89, 90, 91. 
U 


146 Problems. 


ney 
ob 


, yand x belonging to the cycloid, and p, 


becomes at the limiting curve, when for 


and y, x, and y, are substituted; but, od is the tangent — 


of the angle which a tangent line to the curve makes with 
d te 
the axis *, and im or fs is the co-tangent of the same 


) ee are 
angle +, and ra is the co-tangent at the limiting curve: 


d ‘ 
for the same reason a. is the tangent of the angle — 


which a tangent line to the limiting curve makes with the 


: rudd 1 
axis: and since “= — —, or since, the tangent of the 
dt, = Ps 


latter angle equals the co-tangent of the former, one angle 
is the complement of the other: their sum therefore is 
equal to a right angle: consequently, the remaining 
angle of the triangle, or that which the two tangents — 
form, is aright angle; or, the two curves intersect each ' 
other at right angles. [See p. 11.] 


The same conclusion will follow from the expression — 
for the tangent of the difference of two angles |4, B] the © 


hy 
7 


aitmemeietiey penitent 


* Prince Anal, Calc, p. 173. + Trig. p. 9. 


Problems. 147 


denominator of which expression is1 + tan. 4. tan. B*,and 


in the symbols previously employed, 1+), . oe = 0; 
.. the fraction or the tangent (4 — B), is w ; or the differ- 


ence of the two angles is a right angle. 


Pros. 28. 


Suppose now the brachystochrone to lie, not between 
a point and a curve, but between two curves, Required 
the angles in which it must cut them. 


By Form [Aa] p. 89. 
V dx, — V,.8x, + P,.dv, — P,.do, = 0; 
“. as before céx, + P,.dy, — (c.dx, + P,.dy,) = 0, 
and this equation, if there be no relation between the two 
limiting curves, or between x,, y,, and 2,, y,, resolves 
itself into these two, 
c.dx, + P,.dy,=0 [1], and c.dx,+P,.éy, = 0 .. [2] 


If therefore L, (a function of x, y,) =0, be the equa- 
_ tion to the initial limiting curve, by combining it with 
[2] we shall arrive at the same results as were obtained 
in p. 145, by combining M=0 with [1]; since the re- 
‘spective equations are similar. Hence, the brachysto- 
chrone, which here is a cycloid, must cut the first curve 


also at right angles. [See Misc. Taurin. tom. IV. p. 187. ] 


In this last case the origin of the abscissas is supposed 
to be fixed, and consequently, the velocity with which the 


* Woodhouse’s Trigonometry, p. 23. 


148 Probleme. 


body leaves the first curve is variable, varying as ./ Yo If 


the origin of the abscissas is supposed to vary, then, 


[see p. 108.] terms such as —dq ) db a will 


be introduced into the equation of limits: thus, instead 


of the differential of the time being <, , let it be 


Jo) , and let A be a function of a [x,], and b [y,] 
solely ; then, 4 
since, V = v (1 bP) Aa cep ys dh and 

ye (y—h) 2.(y—h)t da’ | 

Vv _ Ja , dh : 

— 2(y—h)? db 


In this case therefore, the equation of limits, at the © 
dh dh 
last point where r=.,, or =a’, since— iq = 2 and = 77 = OF 


[2 not involving a’ and b’| becomes, 
bx. - Pode Onig) [3] 
but, the equation of ae at the first se becomes — 


cd t P,.dy s+ (Se dx ot opty dy JS yd iy 0, [4] | 


Since by the gen’, form*, V — pe a ° ds 


cont) So et rim 
i 2(y—h)? 


and lemawiane — P+ corr. Now at the beginning, 
(y- ape 
the pias = 0, and P yg .*, corr PP. Hence 
=-P—-P=pP — = ee 
fo, @=ht ee oi 2, [from a =" 2. to 7 ae 
the equation [4] therefore becomes 


Problems. | 149 


fe SP, + P,)| ax, + [P, - (P,- P)] 4y, 
= O. 


This last result, expressed in the equations [3] and 
[4], comprehends, as particular cases, the preceding ones 
p- 146, 147; for if h=0, the equations [3] and [4] become, 

c.dx, + P,.dy, = 0, and c.dx, + pani c= 0 
as before ; which equations shew that the cycloid must 
cut the two curves at right angles. This was Lagrange’s 


original determination of the conditions. [See Mise. 
Taurin. tom. II. p. 180. ] 


If h=b, or b—e, [ea constant quantity ;) that is, if 
-y—h, at the point of departure, =0 or =e; or if, in other 
words, the velocity of the body when it quits the first 
curve is either nothing or constant; or, if the origin of 
the abscissas, according to M. Borda’s expression, be sup- 


; Th 

posed to be situated in the first curve: then, = = J, and 
dh 
qq = 0; .*. the equation [4] becomes 

c.dx, + P,.dy, = 0. But the equation [3] is 

feat ee oy. = Os 
consequently, if from the equations L=0, M=0, we 
eliminate, as before p.145, dx,, dx,, &c. there will result 


dy. Cc F dy, 
from the first oe i (ae and from the second aan 


Yo 


d d 
+e P ; consequently, ae = mA , or the tangent at 


150 Problems. 


the point of intersection of the cycloid and first curve, is 
parallel to the tangent, at the intersection of the cycloid 
and second curve. This is the result which M. Borda 
first arrived at [Mem. Acad. des Sciences, 1767, p. 558, | 
and which was afterwards confirmed by Lagrange, and 
Legendre. [Misc. Taurin. tom. 1V. p. 187; and Mem. 
Acad. des Sciences, 1786, p. 30. | 


This completes the solution of the brachystochrone, 


of which there are six cases, [see pp. 113, 122, 141, 145, — 


147, 149.| and this curve which first called the attention 
of mathematicians to the connected doctrine and calculus, 
has been also the object of their latest researches. 


Newton, in the Acta Eruditorum, Mai 1697, p. 223 * 
gave, without proof or the authority of his name, a method 
of describing the cycloid. But John Bernoulli, from the 
Work recognised its author : “ex ungue Leonem” +. 


As additional instances of the use of the equation of 
limits, we will solve the following problems. 


_ * See also Phil. Trans. No. 224. p. 384: his Works by Horsley, 
vol. IV. p. 415: and his Opuscula, vol. I. p. 289. 


+ Quoique l’auteur de cette construction par un exces de modestie 
ne se nomme pas, nous savons pourtant indubitablement par plusieurs 
circumstances que c’est le celebre Newton, et quand meme nous ne Je 
saurions point d’ailleurs, ce seroit assez de le connoitre par un 
echantillon, comme ex ungue Leonem, Joann. Bernoulli, Opera, tom. I. 
p. 197. 


et ee Ti el ee: 


Problems. 151 
Pros. 29. 


Required the conditions for drawing a right line that 
is to be, the shortest distance, between two given curves. 


1. By Prob. 2. p.113, the curve was determined 
to be a straight line; also V = W(1 + p*), and 


i. Se 2a 
<7 + p*)- 


Hence, by the form, p. 89, the pauation of limits 
becomes 
V.dx2, —V 38x, + P,.dw, — P,.de, = 0, 
or, since dw = dy — p.dx [p. 88.], and 


VY. Pr= Wer, we have 


1 
ragyint Pr aa 
a1aeeig) Se 7 e7) 


If x,, x, &c. are independent of each other, this 


dx,+P,.dy, = 0. 


equation resolves itself into 
dv, 


Vv (1 +p’) +P) Aitee o) 


If M=0, and L=0, are the two equations to the two 
limiting curves, there results, as before, p. 145. 


+ P,.dy,=0, and +P,.dy, =0. 


dy, dy. 
———.62,—0y, = 0, Ge dr, — dy, = 0 
dx, f I dx, fe) Q 
‘Hence, by means of these, and of the two preceding 
equations, eliminating d2,, dx, &c. there results 
RSS, iad ence h 
dx, I rR dx, Po 
and consequently, for the reasons assigned in p. 146, the 


line must intersect each curve at right angles. 


152 Problems. 


Pros. 30. 


_ It is required to investigate the conditions for the 
curve, down which, a body, in a resisting medium, 
acquires the greatest velocity. [See p. 138. | 


We have, p. 104, the equation of limits, 

= (VY + aW).dx + [¥’ 4+ (¥’)] x (by — pdx) + 
[7 + (Z')] @xs — pez), 
in which [see p. 140. | 


dz dz 
EF TOS aye 8 tM Oa 
a 2Aep , ; 
=0, (J =~, 2 1, (BH) = A 
\ J/ (1 + p*) i 


Hence, the equation of limits 1s 
dz ; 
o=[F a +A)—2ga+ 2¢rr/(I +p’) | ox + 


im 7H) (dy — pdx) + (1+4a) (4 — =) OX, 


and, by reduction, 
ad 9 


(- — 2g B+ Goto) A Glades 5" dy £(1-+a).82=0, 


and if from p. 139, we substitute the values of 2g and 2A@ 
there results 


— cox + +. (1 + A).dz = 0, 


oy 
Vv a 
which is the general expression: and consequently, the 
two equations at the limits, 2,, z,, become 


— cox, oe, ee + EA $3. == .O, 


- 


Problems. 158 


1 
—C.0X, - Tao: + (1 oe A,) 8%, Pate 
Now z is the square of the velocity ; and if the initial 
velocity be given, dz, = 0; in the second equation, 
since dz, is indeterminate, make (1 + a,) éz, = 0, or 
Aa, =-— 1, and then the two equations are reduced to 


—_ Cub ®. 4. Fa = O, 


— C.0x, + Ft: = 0, 


If L=0, and M=0, are the two equations belonging 
to the limiting curves; then there results, as before, 
p- 145. 


dy, ; 
Tee — dy, = 9, 
dy, | 
og aT oe 


From these two last, and the two preceding, eliminating 
dx, 0y,, &c. there results, 


dy. dy, 
tee Cis a; a c/a, 


consequently, the tangents at the respective points of 


_ intersection of the curve, and the two limiting curves, 


are parallel to each other. 


The conditions relative to the curve of quickest 
descent may be similarly determined. The preceding 
instances are, however, sufficient to explain the use of 
Lagrange’s determinate formule. 


=x ; 


154 Problems. 


The purpose of the present treatise, is now accom- 
plished. There are, however, several curious results 
deducible from Euler’s formula; but, to these, since they 
are not closely connected with the matter of the preced- 
ing pages, the reader is merely referred*. 


* Lagrange, Fonctions Analy. p.217, &c.  Legons sur le Calcul 
des Fonctions, p. 401, &c. Lacroix, Calcul. Diff. &c. p. 660, 667, &c. 
Lexell. Novi Comm. Petrop. tom. XV. p. 127, &c. , 


THE END. 


Published by the same Author : 


THE PRINCIPLES 
OF 


ANALYTICAL CALCULATION, 


1 Vol. 4to. 1803. Price 8s. 


A TREATISE 


ON 


PLANE and SPHERICAL TRIGONOMETRY, 


8vo. 1809. Price 7s. 6d. 


Erratum.—Page 3. line 11. for anounced, read announced. 


es 


10a, a 
aes a * iA 
oon) - “ Bs 


ae 


Pe a es ae 
: ' aa) + ll , 
i dy lich ae, i arr 


cee iy it 
i 
aes 3 yo» '8 
M ian ' 7 
, we we : 
U 


| 


hy in 
ve, 
x 
7 A . 


. : 
’ 
, ‘4 
ny La 
7 r, 
i] 
ay 7 »” 
> : { , 
7 w 
4 ; a 
= 
i 7 


UNIVERSITY OF ILLINOIS-URBANA 


515.64W85T C001 
A TREATISE ON ISOPEIMETRICAL PROBLEMS C 


ANTALYA 


017185551 


