


















Refidual Analyfis; 

A 

NEW BRANCH 

OF THE 

ALGEBRAIC ART, 


Of very extenfive Use, both in Pure Mathematics, 
and Natural Philofophy. 

BOOK I. 

By JOHN L A N D E N. 





LONDON, 

Printed for the Author j and fold by L. Hawes, W. Clarke, 
and R. Collins, at the Red Lion in Pater-nojier Row . 
MDCCLXIV. 












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PREFACE. 


QsV*5 A VIN G t forne time ago , £/> upon a new and eafy 
jjfo met hod of inveftigating the binomial theorem , pro- 

«“€ ' ce f s p^ely algebraical ; / led to confuier whether 
the means which enabled me to invejligate that theorem 
would not be of ufe in the invefigation of other theorems; 
and I foon foundy that a method of computation depending on fuch 
means might be applied in many enquiries. Whereupoiiy believing fuch 
method of computation might be acceptable to the Mathematical World\ 
ly with a view of publifiing it , endeavoured to improve it, and to 
difpofe the feveral articles relating thereto in fuch order as might con¬ 
duce to the eafy attaining a knowledge thereof The refult of my 
endeavours is the following treatife ; in which the elements and com¬ 
mon branches of the analytic art are purpofely omittedy upon the fup- 
poftion that the Reader is previoufly acquainted therewith ; my de/igri 
being only to teach a particular branch of that arty with its applica¬ 
tion to Geometry and Natural Philofophy. Which particular branch 
I have called the Refidual Analyfis; becaufey in all the enquiries 
wherein it is made ufe of the chief means whereby we obtain the 
defired conclufons are fuch quantitieSy and algebraic expreJJionSy as by 
Mathematicians are denominated refiduals. 


The principles of the common Algebra and Geometry having been 
thought inefficient to enable the Analyft to purfue his fpeculations in 
certain branches of fcience j new principles 9 very different from thofe 
before made ufe of have> through a fujpofed necefjityy been introduced 
into Analytics . The Fluxionijls , following Sir Isaac Newton, 

A 2 introduce 


IV 


preface. 

introduce an imaginary motion , and recur to the generation of quan¬ 
tities by a fuppofed flowing, or continual increafe, of their farts. 
Mr. Leibnitz and bis followers, to avoid the Juppoflticm of motion, 
conflder quantities as comfofed of infinitefimal elements ; and rejett 
certain parts of the infinitely fmall increments of quantities as infinitely 
lefs than other parts. In the Refidual Analyfis, (admitting no 
principles but fuch as were anciently received in Algebra and Geo¬ 
metry,) we neither have recourfe to infinitefimals, nor to the principles 
oj motion j but conflder magnitudes as already formed, without any 
regard to their genefis, except in particular cafes, where the manner 
of their being generated may be the proper fubjea of enquiry: And, 
as this Analyfis is not lefs (if not more) ufeful than the fluxionary, 
ot differential, calculus, it will confequently appear, that the Analytic 
art, founded, and carried on, upon fuch principles as were anciently 
received therein, (without the aid of any foreign ones relating to an 
imaginary motion, or irflnitefimals,) is far more exterflve than 
Mathematicians have hitherto reckoned it. 

Quantities infinitely fmall, and quantities infinitely lefs than quan¬ 
tities infinitely little, being i ncomprehenflble ; and the rejecting certain 
quantities as infinitely lefs than other infinitely little quantities, being, 
except in approximations, a very unfatisfadlory (if not erroneous) 
way of getting rid of fuch quantities: the Principles of the method 
of Infimtefimals are certainly liable to fome juft objections, which 
cannot be made to the Principles of Fluxions * ; Jhch infinitely fmall 
quantities as are almofi continually under conflderation in the firfi 
mentioned' method, being no way concerned in the fluxionary doctrine, 
when it is explained and applied in a proper manner. If any thing 
can be faid by way of objection to the fluxionary method, it is, 
that the new principles on which it is founded, though accurate, 
are not the genuine principles of Analytics, for the improvement 
of which, thofe principles were borrowed from the doCtrine of motion: 
And that,- although J'ucb borrowed principles may enable us to give 
very concife folutions to certain problems, yet perhaps we mufi not 
expeCl to bring the Analytic art to its utrnofi perfection, otherwife 

* Dr. Maty, comparing the method of Infinitefimals with that of Fluxions, 
jays, “ Celle-ci a done le merite d’une plus grande exactitude, mais cet avantage 
eit ra chete par ia neceflite qu’elle impoi'e d’avoir recours aux principes du 
mouvement.” Journal Britannique, Moh dc Fevrier 175L. 

than 


V 


PREFACE. 

than by proceeding upon its own proper principles . What weighty 
there may be in fuch objection, I Jkall not take upon me to determine . 
let I mufi confefs, that, how natural foever it may be, in the rejoin - 
tion of certain problems relating to geometrical magnitudes, to confi- 
der fuch magnitudes as generated by motion , it feems to me not 
natural to bring motion into confideration in refolving problems 
purely algebraical: Nor does it feem natural', in refolving problems 
concerning the motion of bodies, to fuperinduce imaginary motions, 
and therewith bring into confideration the velocity of time, the 
velocity of a velocity, &c. nor yet does it appear more natural , in 
the refolution of other problems, to make ufe of the fluxionary method, 
when (as is moft commonly the cafe in that dodlrine) the fluxions 
introduced into the procefs can only in a figurative fenfe be faid to 
be the velocities of increafe of the quantities called their fluents ; fuch 
figurative exprefjion being not the natural language of Analytics, 
but frequently, inflead of conveying clear and diflinbl ideas, is 
confufedly employed in treating of quantities as generated by motion r 
which in reality cannot be conceived to be fo generated *. ‘Therefore 
I am induced to think, that , not only in the refolution of problems 
purely algebraical, but likewife in Geometry and Natural Philofophy, 
when an analytical procefs is requiflte , and what is called the common 
algebra is infufficient, the Refidual Analyfis, which is founded 
(as I conceive) on the genuine principles of Analytics, is, for the 
moft part, more properly applicable than the Fluxionary Analyfis, 
which is founded on new principles borrowed as above mentioned . 
But however, very far from being pofitive in this matter, I freely 
fubmit it to the Judgment of the Public . 

Jn comparing the methods of computation juft now mentioned, 
it may be obferved, that , where the direft method of Fluxions 
might be applied , we, in applying our Analyfis, compute the value 
of the quotient of one refidual divided by another j and, where the 
inverfe method of Fluxions would be applicable , we, from a given 
refidual divifor and a particular value of a certain quotient , 
ajjign the correfpondent dividend, of a particular Refidual form . 
Now, the particular value, which we have occafion to confider 
of fuch quotient , being always to unity , as the fluxion of the Jirft 

* Such quantities are weight, dtnfity, &c. 


member 


n PREFACE. 

member of the dividend to the fluxion of the firfl member of the 
divifor-, it therefore often happens, that, though we proceed upon 
different principles, part of our procefs is the fame as if we bad 
purfued thefluxionary method: So that, in foci, many of the articles 
in this treatife may be of the fame ufe in the dodlrine of Fluxions 
as in this Analyfis ; and probably fome of thofe many articles may 
conduce to the improvement of that dodrine, if the Reader, having 
ftudied it, Jhould chufe to adhere to it. 


The Subscribers to this Work, whofe Names are at 


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- Mr. 


Subscribers Names. 


vm 

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fhall be inferted in a complete Lift, to be printed with the Second Book : 
■—In which Book, the Ufefulnefs of this Analyfis will be exemplified, 
in a variety of very curious Mechanical, and Phyfico-Geometrical, 
Enquiries.. 

Milton , near Peterborough , kANDEKT# 

March 25, 1764. 



THE 

RESIDUAL ANALYSIS. 


CHAP. I. 

Terms and Characters explained . 

T will be proper, in the firft place, to give an 
explanation of certain Terms and Characters , which, 
in the courfe of this work, we /hall have frequent 
occafion to make ufe of:—accordingly, fuch ex¬ 
planation is here premifed. 

I. 

In any procefs, a quantity that is confidered as always retain¬ 
ing the fame value, is called a determinate or invariable quantity. 
And a quantity that is not confiderd as always retaining the fame 
value, but may be taken of any value whatever, or of any value 
between certain limits, (fome other quantity, or quantities, con¬ 
cerned therewith, at the fame time, remaining invariable,) is 
called an indeterminate or variable quantity. 

B An 



w 










2 


4 


THE RESIDUAL 


2 . 


An algebraic expreffion compofed, in any manner, of any 
power or powers of any variable quantity, with any invariable 
coefficients, is called a funBion of that quantity. 


For inftance, a-{-lx™ + cx n and ** * are fundi* 


i* 


ons of the variabl e quan tity x: And, y being equal to any fundi- 

(jxy -i* v* j 

on of x , -- —- may be alfo conlidered as a fundion of x. 

x q_ bVc -f y J 


If y be equal to any funftion of x, it is obvious that x will be equal 
to fome function of y. And if y and z be equal to any functions of 
x, x and y may each be confdered as a function of z ; and x and 
z, each as a function of y. 

3 - 

If, in any given expreffion or fundion of x, wherein x is not 

concerned, x be fubffituted inftead of x , the given expreffion 

and that which refults from fuch fubffitution are called fmilar 
functions of x and x refpedively. 


For inftance, ***«* and 


- - are fimilar 


fundions of x and x refpedively - y the values of *, f g , /, 
and r being independent of the values of x and x : And, y 
and y being fimilar fundions of x and x refpedively, 


ax'y -f- by w ' Sc l -\-xy n may be confidered as a fundion of x, and 
axy -f by m ^c z - f- xf as a fimilar fundion of x ; or the fame 

* > > ii , 

expreffions {ax z y -f- by m s/, c '-\- xy n and ax z y + by m c 1 -J- xy n ) 


may be confidered as limilar fundions of y and y refpedively $ 




Chap. I. ANALYSIS. 

a , b , c, w, and n being determinate, whilft * and x are 
indeterminate. 

4 * 

y being any function of x, and y a fimilar fundion of x ; the 

value of the quotient ofy — y divided by x — at, in the particular 

cafe when * is equal to x, is called the fpecial value of that 

quotient; and * and y are refpeftively named the prime member 
of the divifor , and the prime member of the dividend. 

The faid quotient, which algebraifts commonly denote by 
y — y _ _ 

- or y—y x — x, we (hall, for brevity fake, fometimes 

denote by [x | y] ; and the fpecial value thereof we fhall exprefs 
by [xj-y). 

Moreover, [x _L_y] and [a: ^~y\ being fimilar functions of x 
and x, the fpecial value of [x j^y] — [x j-y\ -r- x x will be de¬ 
noted by [x^.y]: 

And, in like manner, [x"-y] and [x ±Ly] being fimilar func¬ 
tions of * and x, the fpecial value of [x ^-y] — [x u.y] -f- x — x 
/ * ' 

will be denoted by [xjiLy] : &c. 

And the like is to be underftood, when, for any quantities 
having the like relation, other letters or chara&ers are fubftituted 
inftead of x, x, y , and y. 

Accordingly [x j- y] will denote the fpecial value of ^—y -r- 
x — x, y and y being fimilar functions of x and x refpeftively: 
And, Tx-i-vl and [xjlv] being alfo fimilar fun&ions of x 

B 2 and 



4 


THE RESIDUAL 

and x, the fpecial value of \7Ty\ — [* 7J] -f- ~-^x will be 
exprelfed by [xny ]: &c. 

So [y j, z ] will fignify the (pedal value of z — z -f- y _y, 

z and z being fimilar functions ofy and y refpedively: 

A ad [y-Lz] and [ yj.z] being alfo fimilar functions of_y indy, 
[>' — z ] will exprefs the fpecial value of [_y^z] — Lv-^] — 
y — y : &c. 

5 * 

Let it be noted, that, whenever y denotes any fundion of x; 
y> y* y> y> (hall denote fimilar functions of a:, x , *, *, &c. 

’ " / •• m 

refpedively. 

Likewife, whenever z denotes any fundion of y ; z , z } z , z, 

/ " m 

6cc. (hall denote fimilar fundions of y, y , y, y, &c. refpedively. 
And the like is to be obferved, with refped to other quantities. 

6 . 

When any quantities v, x , y, z , &c. are concerned in any 
procefs, and the quantities v, x , y, &c. or [ti A x], [v^y], &c. 

are alfo concerned in the fame procefs ; it mud be underdood, 
that x , y, &c. are each confider’d as fome fundion of v , though 
their being fo related be not there particularly mentioned 




THE 


Chap. II. 


ANALYSIS. 


5 






RESIDUAL ANALYSIS. 


CHAP. II. 

Of the Invention of Rules neceffary to faciliate Com¬ 
putations in this Analyfis . 

IT323£"*3N making computations by the method taught in 
KV *o[ ^* s Treatife, wc &all frequently have occafion to 
£> 1 5J3 aflign the quotient of F—F divided by x — x 9 

F heing fome fun&ion of x or, at leaft, we fhall 
often find it necefiary to have recourfe to fbme 
Rule derived from a confideration of the relation between the 
fpecial value of fuch quotient and the prime member of the 
eorrefpondent dividend. Therefore, as the Books of Algebra 
hitherto publifhed are deficient in refpect to fuch divifion, it will 
here be proper to- fhew how fuch quotient may always be 
obtained ; and to deduce therefrom fuch Rules as may be ufeful 
to facilitate computations in the fucceeding Chapters. 


The Theorems which chiefly enable us to perform the divi¬ 
fion above mentioned are the following, viz. ^ w 


v r — Ttr r 




6 


THE RESIDUAL 


V r — w r 
V — W 


v m ~ l 4- v m ~~ i u> + v m ~hv\ J r v m ~ A w* (n 


iw m 3*» zm 4 m 3 m 

4" V ryjrJ^.’i) r W r 4“^ r W r (r) 


+¥ +¥ w 

—l” « _Vn 

i + =b.fSb + =Wr) 

<V I «ll <v I 


m and r being pofitive integers ; 
and uw — uw = wx« — u u \w — w ; 
which, by an eafy multiplication, will be found to be true. 

m m 

From the firft equation it follows, that -— r - ^Z . w —- — — 


• is = — v w r x 


vwv X v — w 



+U‘ +¥ w 

—1 

m 

zm 3 m 

1 +^ 

•V 1 




* This theorem may be inveftigated as follows. 

It is well known, that 

m m 

~ v~^ ' w is = ' r ~ t + («), 

a „d / ~ f =«'-’ + a r ~'b + a r ~H' M, 
w and r being pofitive Integers. 

m m 

In the fecond equation write v r and w r inftead of a and b refpeftivelv, 
and you will have 

mm m zm m zm zm 

V — W m— — m -— « — A- — 

- ~m= ZV + V r W r +V r ivr (r). 

v r — u> r 

Then, from the firft and third equations, it will appear by divifion, that 

m m 

— — m — 1 m — 2 , , 

V T — IV r . <u 4 . V W (tn) 

-IS —---: 

V - W m m_ 

v m -- 7 +v n r w r (r) 

Which being fo obvious, it is matter of furprize to me, that Algebraifts have 
not before obferved it, and (hewn its ftngular ufe in Analytics. 

Example 



7 


Chap. II. 


ANALYSIS. 


Example I. Taking — = ^ we have 

I_i »+=+=]*+3' 


± 
V 3 < 


•+W+V 

Whence it is evident, that, when iv is = v, the quotient of 
^ 3 — w 4 divided by -u — w is equal to 3 . 

Moreover, — being = 1.3333 &c - 


v 3 —■ w 3 -f-'i;—<zy is = -y s x 


or, more nearly, = -y 3 x 


• 3333 &c - 


+¥ + ! 

] ('3333) 

*+l f +#+^ 

| 3 (IOOOO) 

-+i +1T+1 

J 

( 3 33333) 


nearly; 


—V±. -,_± —,** 

*+ir r +i 3 +ii T 

&c. 


(100000) 


Hence it is likewife evident, that, when w is = «y, the quo- 

4 ± ± 
tient of , y 3 — w 3 divided by v — w is equal to —u 3 : For, the 

ratio of 1333 &c. to 1000 Gfr. being as 1.333 & c - to *> 

the ultimate value of l±l ~j~ 1 + 1 ( 131 L *£-1 (the value of 
i + i+ i+ i (1000 See.) '‘ Cne Vaiue or 

. nv , oul 

1 + « +tI (*333 &c.) 

- when w is equal to v) is manifeftly 


* + 


3 s + 3 Ii 

<1/1 * <i/i 


3 (1000 &c.) 

equal to ±, the quantity from which (by divifion) 1.333 is 
obtained. 


Example 


4 


8 


THE RESIDUAL 

Example II. Taking ^ = V 2 = 14142 we have 

—=-r= .+= + = 1 ‘ + =f ( 1414 ). 

v - W -—V - Wz=V x-7=-7=-7^-g. 

v w ^ 2 -2 -1 3 ^* v< 

*+=| +j\ +| <‘T 

^7_. »+= + 3 L + - 1 ' (l4,42) 

or, more nearly, == v x —7^ ' 


—V r —I 1 '' X * 

'+=1 +1 +1 ( '° 


CST. 

Whence it is evident, that, when iv is — v, the^quotient of 

,/*_ </ J divided by v-w is equal to Si x v 2 ~‘: For 
the ratio of 14142 Vc. .0 10000 £fc. being £14^ 
is manifeft, that the ultimate value of x x _±_ 1 (ioooo tic.) 

, +1 +1' (*4142 &c.) 

. t . c ' v ~ <r " _ Z ___ when w is 

(the value of - . r . 

i+l +1 +t! i* 0000 * 0 

equal to v) is equal to s/2, the quantity from which (by ex- 
trading the root) 1.4142 &c. is derived. 

,+= +1‘ + =f <•>. 

_ r . y _ Z ___is equal 

Corollary I. Seeing —*; UTE—t 3 - , s 

.+H' +!'+!' (0 

to £ when « is = Vi it is evident, that the fpecial value of 
r> 

7 = 3 'is equal to V whether £ be pofitive or 

Hence, by fubllituting f, x, and X, inftead of v, and it 
refpedlively ; it appears, that,^eing^either pofitive or negative, 
(the fpecial value of x ! f~r’ x * Coroliary 


Chap. II. 


ANALYSIS. 


Corollary II. k-\-av r -\-bv’ &c.— k no' -f- bw ‘ &c. 

mm n n 

being -=. a x v T — w r b xv * — w 1 See. 


av r - bv' See. — aw r bw ' See. -4- v —- w 

m n 

is =z av r 1 x M + ^ * * x N &c. 


and av r -f- bv ' &c. — aw r bw ' &c. -f- v — w 

m 

r x M — bv w 

i-f- 4*^1 +^T 

v ' «yi -v 1 


= — av l w r x M — bv 1 w 4 x N &c. 


M being =: 


■+=r + a r +=i 


w 


w 


and N = 

&c. 


■+S +1*+1‘ w 


w 


£s?r. 


Moreover, 


[*.l*+ «/+£** &c.] is, in general, = paJ’~'-\-gSx , ~'&c. 

2 . 

w m 

Suppofe z 7 — z 7 -f- — F, and — f: 

Then, z—z being =f x — f x> z r — z T fx — f x will 
be = F, and z r — z r -f- x — a; = Ff 

Example. Let be fuppofed = ax -f * 7 , and s = ax 4 - *7 : 

‘ / 1 / 

then, 


THE RESIDUAL 

' ¥ 

1 + 7 + -| M 

then, F being (by what is faid above) = z r x- ^^— 5 

¥ Ip 

i + - + d (0 


and f — a x 3 x- 


2 r -f- * — * 


1 + 71 + 
will be 


1 + t + 

= z r x- 


►T c 


- iHL —.if! 


X a + 3 X • 


* 3 

* + t +■ 


Corollary. When a; is = x, z will be =zz, and f =[x ^-z], 
/ • 

z being any function of x ; moreover, by Cor. i. of the preceding 
Article, F will be f being written inftead of 

Therefore [x ^ z will be = pz P ~~ 1 x [x z]. 

Example I. Suppofe z = a -f- & c * Then, 

by the preceding Article, being = bmx 1 + 

cnx~~ 1 &c. [x _i. a + bx™ -f- ex &c.| J will be = 

p x a -f b™ -f- cx &c.| x bmx 1 1 cn * * 

Suppofing the coefficients c , d y &c. each = o, we have 

[x<7 + bx‘ 1 J = bmp x + x x™ \ 


Example 


Chap. II. ANALYSIS. 

n '- W 

Example II. Let z be fuppofed = cx a bx 


Then cx + a + bx™\ — ex' + a + bx”\ being = c . x — x 
-\- a + bx m t •— a + bx\ > [x cx -J- a 4- bx’°f 3 will be — 
[x_'_fx”]4-[*—^ bx l | 1 cnx 1 ~ 1 bmp x abx 


and 


[x -j— cx -f- ci + bx m \ | J = q x cx 4 * a 4 ~ bx™ x 


XX 

[? I 


t> — I m — I 


cnx 1 4" bmp x a 4- bx\ x x 

3 __ 

Seeing uw — uw is = w x u — u 4-« X w — w, as we have 


before obferved ; it mud: follow, that uw — uw -f- *v — v is 

U - U w — w * 9 * 

— W x -- + u x- 

V — V 1 t v V 

Example. By taking u , w, u , and w refpe&ively equal 


to a — v 1 ] , a x 4- v*y a — "*y*| , and aS t we have 


*v x 1 x ax 4” — u — t u i | x #‘ 4” —f- *y — = 

v -|- v 


— C? -j- v { X 


— v z 4- vV 


== + vV — 1 - x —I-- ; 

a —v* ' 4 - 


a — v‘f — a — v'[ -f- v — v (by what is faid above) being 
V 4- v _ 

equal to — v = T =—^=== %> and a 1 +«*- a' + f 
hj — v v z — v* -4- v — v) equal to 1 — j. 

* / / v 1 -f- v l 

C 2 


I I 


CoROL- 


12 


THE RESIDUAL 

Corollary. When v is = v, u and iv will be equal to u 

and w refpedively, thefe laft quantities being any fundions of v. 
It is manifeft therefore, that 

['u u r ijj\ will be = w \y j- ii\ -f- u [*u w\r 

Example I. Suppofing u and w to be refpedively equal to 
x m and y n , (i. e. fuppofing x and y to be equal to any fundions 
of v 9 ) we have [v x m y n ] = f x [v^ x m ] x m x [vjl/i 

= mx m l y n [«u x] -f- nx m y l 1 [v _i_ y]. 

Exa mple II. Suppofing v, u , and w equal to x , cx m , and 
a -|- bx r \ n refpedively, we have 

[xj_cx m xa + bx r \”]=a + bx r fx [x^cx^-^-cx^x [*^_zz-f-Ax ; f J 
= cm x a -f- bx 1 x x m ~~ l + cnx m x a + bx' I x [* _i_ zz + Av r ] 
= x zz-f- bx r \ x 1 -j- benr xa-\- btf\ xx"^ r x . 

Example III. Taking u and w refpedively equal to a-^bx^ 
and c-\-dx r i , we find 

\y -i— x c dx r | ] = c dx r \ x \y __l zz -)- J 

+ zz -J- <$x”| x [v jl c + dx r \ J — m x c -[- dx r \ x zz -)- £v”| 

x [*y -i. z? -j- Ax” J — zz 4" Ax”l x c + dx*\ x[v -i-c -j- dx r 1 = 

bmn x c -\-dx r I x zz -j- bx n \ xx n ~ r [ r J _i. x] — dr x zz -J- bx tl \ x 

x* r 1 fu x] ; zz, A, c , z/, //z, zz, and r being de¬ 
terminate, whilft x is indeterminate. 

Example IV. y being any fundion of x> we have 

[x j-abx n \ X f~\ =y r x [x _jl zz-j-Ax”l ] + zz-j-Ax”l x [x_i.y r ] 

~my r x a-\‘bx n I x[xj_zz-}-Ax”] + ry r ” ^zz-j-^1 x[xj_y] 

•= bmnx n ~~ J y r x a-\-bx n i zj'"" 1 x zz -f- Ax" l x [x _',y]. 

Example 


*3 


Chap. II. ANALYSIS. 

Example V. 

[x j. z [xz)] is =[*.-.2] 4 * z [* — z], 
[x-uz[x^.z-]] =[x^2]x[xjl2]+2[x^2], 

[x ^.2]] == (* — .)’] x [^ — 2] Ar y [* —^], and 

[* [x ^_y] x [x 2]] = [x ^.y] x [x _i. z] -f [x -L.y] x [x^.2]. 
y and 2 being any fun<Sions of x. 


4 - 

uw — uiu being (as is obferved in the laft Article) =®x »—u 

-f- u x w— w = A 5 uwx — uwx will be = x A + uw xx — x 

/ / / / / / / 

= ivx X u—u xu x w — w uw xx — x r= B ; and 

• • / / / / 

uwxy — uwxy r=jyB -}-uwx Xy —y — wxy xu — u-\-xyu X w — w 
-f* yuw xx — x -}- uwx xy —y — C ; &c. 

11 / tit / 

It is evident therefore, that uwx («) — uwx («) is == wx {n — i) 

xu — u -\- xy(n — 2) x u x w — w -\- yz (n — 3) X uw X x — x 
-r («) J a nd, confequently, 

__ _ u — u 

uwx ( n) — uwx ( n) v — v — wx (?i — 1) x + xy(n — 2) 

W - iv X - X 

x "Xi~;+;■*(* — 3 )X'/®X~+ (»)• 

' / 

Example. Taking u, w , //, <zr, and * equal to v T , 

__I_ _ ^ I_ __ , _ 2 

+ *^1 *, ^ “1“ 5 j < u } , -|- v\ *, and b v \ 1 , refpe&ively ; 

. JL --t . \ I I . z 

we have ^ 1 x a -|- v { 1 x ^ 4 * V 1 ; — ^ ? X a 1 x /» ^7 



’* + 


THE RESIDUAL 


-f- v — v = a-\-v T xb + ^1* X v 1 x- 


£ + vl* X'u 1 X+ vl 4 X- 


* + ■ 


b -f- 'ul 3 x 




b + v 
b -f- ‘v 


Tli rl‘ 

1 +^i + *i 

-)- v~ xa -)- v \* x 


T + 


b -f- «i/\ T £ -f" *• 

1 b-^-v' £-f-«lJ 

is faid above) being equal to v ' X 


.. — V i —v (by-what 


vh "•yl f 

+ + 


a — a -f- v\ * v — v equal to a-{-v 1 * x - 


i + 


a -j- <v ' 
+ v 


and b -f- v \ J — b -j- v \ 1 «-f- v — v equal to 

b •v 


b v i 1 x- 


1 + 


b <v 


1 + 


b +; 

b + v\ 


■ jqr 


Corollary. When ^ is equal to * 17 ; u y w, x, y, &c. will 

be refpeftively equal to u , ,w, *, _y, &c. thefe laft quantities 
being any functions of v. It is evident therefore, that 

[v jl uwx '(«)] will be = wxy ( n — i) x [-u ii\ + uxy {n — i) x 
[ ujlw ] -f- uwy{n —i) x [‘Uj.a:] + ( n ). 

Example. Writing r J > , w 9 , x r , y\ inftead of u, w, x y and 
y refpedively: we have J/u _i if^x/\ = v?x y x [ < u^_ /J + 


T s 


Chap. II. ANALYSIS. 

u xy x [v _£_ w qm \ ifw q y x [v -l. * r ] + if'-Jx x [v jl /] = 
pw q x r yif ~*x [v-i. u] + qu xyv?ru r Jyx 

X [v j- x] -f* suw , xy~ ml x [‘U -l.^] j u, iv, Xy and y being 
any fundtions of v. 


Suppofe F — F -f- x — x y = Q, and /—/ -r *— * = q : 
Then, F — F being = Q.X a, — and /—f = qxx — x f 
F — F -r* f — f is manifeftly = —. 


Example. If F, /, F, and / be equal to ax + a; at 1 , 


ax x 1 ]^ and refpedtively j F — F -r- /—f will, by 
what is done above, be 


ax -f- x 1 


ax x] J x 


i + ; 


ax **1 < ax 4 - jF\ * 


1 + 


-F **1 


4 - ^ 


TX^ + AT'r^*-r- 


x~ + X 


Corollary. F and f being any fundtions of x, the value 
F — F 

of the quotient or fradtion yzzj' when x is = (*■ e • when 

both numerator and denominator vanifh together,) is = . 

5 i*+n 


Example, 


THE RESIDUAL 


16 


Example. If F and f be equal to a 1 and a -f- x\* 

F — F f\ 

refpe&ively, the value of ~I1J ( = —-I- —— 3 J , when 


— a + x\ 


x is = x, is equal to 


3 xXa 1 **1 


£+Zr 

« + * • 


6 . 

N and D being algebraic expreflions fo compofed of x and 
other quantities, that each of thofe expreflions becomes equal to 
Nothing, when x is equal to fome certain quantity k ; it is 
obvious, that N and D (the expreflions which refult by writing 

x inftead of x in N and D refpe&ively) will both vanilh when ue 

is equal to k. Therefore, if, in the quotient of N — N divided 

by D — D, x be taken equal to k , the refulting expreflion will 

be equal to let x be what it will. Now, by what is 
fhewn above, the quotient of N — N divided by D — D 
N — N x — at • 

(— ' __1 i mav always be readily afligned in terms 

D — L> -j- *■ — x j 

which lhall not vanifli when x and x are each equal to 

unlefs the value of that quotient be then = o: confequently, 
by afligning fuch quotient, the value of ^ will be had, as well 

when N and D are each equal to Nothing , as in any other 
cafe. 


ExAxMPLE 


Chap. II. 


ANALYSIS. 


*7 


Example I. To divide s/za + 2x 1 — 2 a’x* by x — a. 

The dividend and divifor both vanifhing when x is equal to 
a, it is obvious, that, if, in the quotient of 

✓ 20 1 -f- 2 x % — 2 a T x T — ✓ 2 a* -}- 2 x' — 2a T x T divided by 


* — a — x — a, a: be taken equal to a, the refulting expreffion 
will be equal to ^ 2a , let x be what it will. 

x — a 


✓ 2 a x -f- 2 x x — 2a T * T — V' 2 a % 4- 2jr l — 2*** T 

Now *--- - —--is manifeftly 


2 a x 2 x x — V 2 a x 4- 2 x x 



X - AT 


2 a* x 



; which, by what 


2 x .*• 4- *• *. 

is done above, is = - - -- — -. 

V 2 a x 4- 2A * 1 4- V 2a x 4 - 2x z 

Confequently, by taking x equal to a, we have 

•d 2 a x 4 - 2x x — 2 Jx * 2 x 7+7 _ 2 a* 

2 a 4 - V 2 a' 4 - ix x a \ a \ x h jj.t 

Whence it is evident, that, when x is equal to a , the quotient 

of s/2a -j- 2x" — 2a 1 x T -4- x — a is = -. 

3 

Example II. Suppofe N and D equal to s/ik'x — x + — 
\/kx' and 2X — 7? + refpeftively. 






18 THE RESIDUAL 

_ _ _ _2^ a — x 3 —• *** — xx % — x l , 

\//bc ; — v/ kx ' -f- X-Ar= ' — k T x 7 X 


— ** 4- y/2&*x - a 


^l£l 

jf 

I + - 

X 


7 -—> and 2 * — 7 ^ -f- * 3 [ J — 2 * — 7 ^ + * 




* — * = 2X 2X -f- a: — X - jk 5 -f- x’l* 7 ^' 

a_ Jf* -J— A-Jf -|- AT X 

-r- a:- * == 2 — 7 ^ -j- x 5 ) * X- 


1 + 


7 * , + * , | I 7 * J + f I 1 


N — N -f- AT — ; 


7 *J + *“1 + 7 b+ X i\ 


D-D -r- x — x 


-1 will be =■ 


2t’ 


• 71 

— at 3 — ArV - ACAC*- AT 3 j 4 ._I_ 

------ — k T x T X ---r^- "Zr 

■4 


y/ 2 & 3 x — X 4 -4- s/ 2Px — x 4 
* / 


X* -f- AfAf X* 


2 - 


7*’ + *’(' + 7*’ + **| T X7 ** + f’l T + 7*’ + * ! l' 

From whence, by taking x equal to i, we get yj = 

x*—h'— P* T-,i 1 + 4 + 4 

- . - S). x - 


l/2^ 3 AT-AT 4 -}* 


■ + 3 ‘ 


2 — 


Af a ^ ~b 


7 <’+*’l i + a* * 7 *M- 4 V + 4 >‘ 


Hence 


*9 


Chap. II. ANALYSIS. 

Hence it is evident, that when x is === k, 
l _ */7r 


s/lk'x 


10 , 

■ is =- k. 

7 


2x — jl* -f- at 5 j 

Corollary. Seeing it amounts to the fame thing, to take 
x equal to #, and afterwards * equal to t; as to take, at firft, 

x and x each equal to k: it is obvious, that the quotients or 

fra&ions re and ^ become equal to each other, when in 
O DJ 

each, is taken equal to k. 

Example I. Suppofing N and D equal to s/ 2 r x — x* — 

. - i_ 

y/rx J and 2x — yr % x *1 1 refpedtively j we have [x _i_ N] = 
r \-^- l r r x r t and [x ^ D ]= 2 -It 

appears then, that the value of ^ when x is taken equal 

to r, is = — 121. and the fame is the value of iL when x is 
7 V 

fo taken; which agrees with what is done in the preceding 
Example. 

Example II. If it be required to find the value of the 
quotient of uy/ i — u — us /1 — u divided by u — «, when 

t 

u is therein taken equal to u ; we are to confider u as inva¬ 
riable then, the fpecial value of 


us /1 — U us/1 u — us /1 — ll — uV I 


u — u being = • 




— s/1 — u and u—u — u—u 


u — u being = — I, we fhall have 


y/i — i 


s /1 — u 


(= vT ^-aO ^ or ^ ie required value of us /1 — u 1 — us/1 — u 

D j2 Example 


-f- h — u. 









20 


* 


THE RESIDUAL 

Example III. Let s f c , and y be any functions of u ; anft 
let it be required to find the value of the quotient of 

sy — yxcus/i —u — cus/ i—u -f- suu + s/i -« l x /\ — u 

divided by u — u, when u is therein taken equal to it. 

Here we are to confider u , r, and ^ as invariable 5 and 
we may obferve, that the propofed quotient is 

= 5 * y — uuy — y /1 — u x y/i — u -r* u — u — 

• 1 • / / 

fy x us/ 1 — u — u /1 — u -4- u — u. Now the fpecial va- 
lueof^y— uuy — y/i-—/x/ 1 -V— y—uuy —y/i-u x /\-u 

•I ’ / "V/ 7/ // 

u — u is = — [u^.y\x uu + >/1 — u x v^i — a‘ — 

1 — u x ■ - — — _ 

u y + ■ > and u — u — u — u u — « is = — 1 * 

VI — u x ' " t tr 

moreover, the value of us /1 — u — u/i — u -4- u — 

when u is = u } is equal to —_ by the preceding Example : 

therefore, [u ^.jy] being = [u^y] when u is = #, and y being 

then = y; s [u _l y] — ^ ? - - is the required value of the 

-quotient propofed. 


Example IV. Suppofe y to be any function of x: 


y — y 


Then, [x 4. y] — [a: | = [x j. y] — ^ - being = 

7^x[x-Ly]—y-y * —f x [x ±y ] -~y 

- ---, we have-- ~ 


where 



Chap. II. ANALYSIS. 

where both numerator and denominator vanifh, when * is = * 
(y being then = y). 

Therefore, by our rule, the quotients or fra&ions 1 ^ 

— l*-y~\ + 

and-=====— are equal to each other, when x , in 

— 2 X X — X / 

each, is taken equal to x . But both numerator and denomi- 
nator of the fraction--vanilh, when x is equal 

— 2 X .v — x , * 

i 

to *. Therefore, applying our rule a fecond time, it appears 

O—XI — O I y] [x-^-y] 

that the quotients or fractions —-and -—are 

X —’ x 2 

equal to each other, when x , in each, is taken equal to x . 
Confequently the value of the quotient or fra&ion ^ x ~ y ^ 

i 

when x is taken = *, is equal to which, it is 

. / [?—y] 

plain, is the value of —when x is taken as juft now men¬ 
tioned. 

7 - 

y, z , &c. being any functions of x; and F and f being 
algebraic expreflions compofed, in any manner, of all, or any 
of, the quantities x> y , z , &c. and any invariable quantities : 

If F be always — f y 
F will be = f y 

F ~ F =/-/> 

F—F * =f—f -i- a;—*, 

and_confequently [.v^F] = [x 


t$* The 



22 


THE RESIDUAL 

& The deducing this laft equation from the firft, (and like- 
wife every fimilar operation,) I call rejtdnal dirtfwn :—For the 
k ready performing of which, the Corollaries to the firfl four 

Articles may ferve as Rules. 

Example I. If ax be = by 1 , we fhall, by refidual divifi- 
on, have 

a = [x j. by*'] — bmy 71 1 x [x _l y] ; 
or a x [y + x] = [y by m ] = bmy m ~~ l ; 
or ax[v -i. x] = [v by m ] = bmy'^'x [v J~y\> 
v being any function of x or y. 

Example II. If y be = a -f- bx — cz , we (hall have 

[* - y] = b — fx[x-^4 

or i = b [y _i_ a:] — f [y _l z], 

or [2 -i- = b [z *] — f. 

Example III. From the equation ay m = we get 

1 [x j-y] = — 2wx — *f 1 ; 

or amy vn ’“ l -=— X^ a —.* l |* 

or amy n ~~ 1 x [v j~y] = — 2 nx [*j x] X b l — ~ *, 

v being any function of x or y. 

Example IV. From the equation x = ay 1 z , we find 
1 = amz y x [x j- y]any z x\xj-z\. 

Example V. If * be *= Z + \/y l -f z = yz~ l + 
y 1 2*| t , we fhall find, by our refidual divifion, 

. __ [x^y] ?!>—*] . *] + *[* — *] 

“ ~ + ftW 


Example 


ANALYSIS. 


2 3 


Chap. II. 

Example VI. a ' -f- -f being = x p x a -\-x n ^, we get 

m —I r t — i n fll^ n -X- p — I n 

tny [xj-y]—p x r Xtf+x| 

Example VII. From the equation ax'y -f- bxz 1 = cy'z\ 
we get 

2 axy + ax 1 [x _l y] + + zbxz[x ^ z] = icyz * [.v .^y] -f- 

a?]. 

Example VIII. ax m -\-bx?y q cy n being = o,. we have 

amx 1 1 bpy q x^ 1 -f- bqx^y 1 1 [x_i_y] f»y” 1 [x _i_y] = o. 

Example IX. Suppofe a; = ay 1 [x s~y] : then, by our di- 
vifion, we find 

i 

i = 2 ay [x y] ay % [x jjl y ], 

Example X. Let axy + bx % [x y ] be fuppofed = o: then 
we (hall have 

ay ax [x _i,y] -f- 2 bx [x ^ y ] -f- £x* [x .".y] = o. 

X 

Example XI. *x[x_*_ z] by [x ^.y] -J- c\xj-y] being 
= o, we find 

a [x z] -{- ax [x ji. z] -j- b [x _l y] [x JLy] -f- by [x ^ y] -f* 
2c[x^y] [x «_ y] == o. 

Example XII. Suppofe x* + axy * 4 * by = o: then, by the 
divifion fo often mentioned, we have 

2x -f- <*y -f- ax [x -i_y] -f- 2 ^y [x _i_y] =- o : 

And from hence, by a fimilar procefs, we get 

2 ' 2a [x -*-y] + ax [x _£Ly] -|- 2^[x ^.y] 2by [x Jiy] = o. 

Example XIII. Ify be = tfx™ j [x ^.y] will be = amx m *, 
[x^y] — am . m —i. x™ ” 2 , and [x _«.y] z= am. m — i. w —2 

. x m 3 , Gff. as appears by the method of divifion above-men¬ 
tioned. 8, 



24 


THE RESIDUAL 


-is manifeftly =-j-: it is obvious therefore, 

X — X '■ y—y y—y 

0 9 9 

that, y and z being any fun&ions of 
[X-LZ] is = 


y—y * — x 

ZZT X X jz~y bein S = *» [ x — y]x[y — x ] is evidently = i, 

and ["* — yl =--—. 

_ .. . J l*-*-*-] 

From this laft equation, we get, by refidual divifion, 

[x jLy] = [a; ~ t which (by what is (hewn in the pre¬ 
ceding Article) is = [y —-—1 x —-— = — ^ f'.il 

Moreover, from the equation [x y] = — we 

ly-i-x] 

find, by refidual divifion, [x ji;] = — [* j_ *\ "I = 

l>—*] J 

— Tv , l v _L__ 1*^*1 , 3[ y-e-*] 1 «. 

LJ '-[y^x ] ’J x [ y^x]--iyT^ + rr^F’ &c - 

From what is done in this Chapter, it appears how the values 
of the quantities [*_2i_y], &c. or [y _*_#], [y x], &c. 

may be obtained in terms of x and y } from an equation (hewing 
the relation of the two quantities laft mentioned, when no expo¬ 
nential quantity is concerned therein ; which values we (hall 
have frequent occafion to compute in the inveftigation of propo- 
fitions by this Analyfis. Rules for computing the values of the 
faid quantities [x ^y], [x JLy], &c. [y _i- x], [y jjl *], &c. from 
equations containing exponentials, will be given in the next 
Chapter. 


THE 



Chap. III. 


ANALYSIS. 


25 


| aoooooooc 1 

THE 

RESIDUAL ANALYSIS. 


CHAP. HI. 

Of Expo?7entials and Logarithms . 



S one variable quantity may be denoted by the 

algebraic exprefhon x”, or the like, where the Root 
only is variable whilft the Exponent remains inva- 
variable ; fo may another variable quantity be 


denoted by n , or the like, where the Exponent only is variable 
whilft the Root remains invariable. Therefore, having fhewn 

.how x — a”, may be divided by x — x , it will now be proper 


to {hew how n — ;/* may alfo be divided by the fame divifor 
(x — x ).—In doing which, we fhall firft aflign the value of n 

in a certain feries of terms of n and „v, wherein the exponents of 
the feveral powers of thefe quantities fliall be invariable : by 
help of which feries we (hall be enabled readily to obtain the 

defired quotient of n —- n* divided by * — x. 


£ 


We 


26 


THE RESIDUAL 

We (hall then (hew how, by means of that quotient, the value 

of n may be afligned in another feries of terms of n and x , in 
which the exponents (hall be invariable and likewife how, from 

the equation n = *u, the value of x may be afiigned in a feries 
of terms of n and v, with invariable exponents.—From which 
feries many ufeful conclulions relating to Exponentials and 
Logarithms will be eaiily deduced. 


I. 

ID 

_ ___* __ I 

Affuming x r = i+b.x — i + c.x —1| d» x — i| 
&c. it is propofed to find, in terms of m and r, the coefficients 
b, c, d, &c. 

Writing (in the affumed equation) y inftead of x , we have 

y r = i + — I -f- f • y — J l + d. y — 71 &c. 

and by fubtra&ion, 



If, now, we divide by the refiduai x— y, we (hall get 

‘+i +H (•> _ 

’+T1 t +T1 t w 

___ _ - - _ i _____ _— _x 

h^-c.x —-i+y— i-\-d.x —71 -}-x—i xy — i -\-y — i] 
&c. which equation mud hold true let y be what it will : 
From whence, by taking y equal to x , we find 

— —, 2 

~ x .v r = b -f- 2C . x — i + 2>d , x — 1 1 &c. 


Confa- 



Chap. III. ANALYSIS. 

Confequently, multiplying one fide by x and the other by 


3 *— i, we have - x * r , or its equal 


■ i &c. 


™ _L ™b t x j J- ™c . x il 4 - ~d . x 

■r ' r ' r 1 r 

= b + + ^d\ x '— T 1 ®* 

Hence, by equating the homologous terms, it appears, that 
:b is = ^, c = d =— ,c, 

r 2 r y 4 r 

In like manner, if we aflurne 

= i b, x — i + c • x — i|‘+ d. x — 1 1 &c. 
x r 

we lhall find b = — c = — ?L±I. b, d = — c, &c. 

r 2r y 

Confequently, —— being = x~~~, it is manifeft, that, 

x r 

whether the exponent — be pofitive or negative, x~ is = 

+ Ti* + 2 . 

&C. 


m m—r m — lr 


.X — Il 


Hence, by writing n indead of x , and x indead of ^, we have 

i 2 ' 1 2 3 1 

&c. a: being any number whatever. 

And, by fubftituting, in this lad: equation, inftead of 

7 i y it appears that az\* is = a* 4 - x.a* - J z 4 -x . — . a x " 

a 1 . -—-. -—- . &c. which is the Binomial Theo- 

2 3 

rem *. 2. The 


27 


* The fame conclufion, by referring to what is proved in the preceding 
Chapter, may be obtained in the following brief manner j but I prefumed, 

£ 2 that. 




28 


THE RESIDUAL 


The expreffion ”—— 1 . being = n*x ” ' ~ r , which by the 

* x x — X j 


preceding Article is 


X — X — I X — X — 2 


== ,/x «- 1 + —— + n* 

Gfc. it is obvious, that, when x is equal to x, the value of the 
quotient of n —— n ♦ divided by x —— # is 

— * x;2 1 -$-r -y--— & c.=p« , 

£■ being put for the feries n — i — 7 ~~ + ^- 

which converges when » is a pofitive number lefs than 2. 

Writing i. inftead of n t we get 

fpec, val. of -7 -f- x — * =-— x fpec. val. of 


: — X 1 ---— . - - 

_■* n 2 n 


n , I I — 

‘ 3 n~ 


tkat, upon the firft application of what is there taught, it would not be amifs to 
give a more explicit procefs. 

The brief inveftigation is this. 

Afluming 1 4. — 1 4- bv 4- cv' -}- 

we get, by refidual divifion, [y —• v being the divifor,) 

x x 1 -f- =: b + 2 cu -f- &c. 

Confequently, multiplying by 1 -f v, we have 

x x 1 -p v I , or its equal * -f- 4" 4“ *^ 

= * + £h*+ 

From whence, by equating the homologous terms, £, c , &c. will be found, 

equal to *, - & c ‘ re ^P e< ^ lve ly : And then the value 

of. a -f z|. may-be obtained by fubftituting ^ inftead of sv From 



29 


Chap. III. 


ANALYSIS. 


From whence we find, that the fpec . val. of n — n* -f- x — * 


is alfo = n x‘ 


■ + T~ +4' 


&c. = £ x 


g being now put for the feries — ^ ■ 4- 

converges when « is any pofitive number greater than 


Affuming n x = i Ax 4 - Bx 1 4 - Cx 1 See. it is propofed to 
fnd t in terms of n, the coefficients A, B, C, &c. 

ri* being = i 4- Aa: 4 “ B# z 4- Ca: 5 See. 

we have, by fubtra&ion, 

n — 7 i' = A . x — a: 4 “ B . a" — x z 4- C. x' — x 1 &c. 

/ / / 

and from hence, by divifion, 

X x __ 

n = A + B . x 4- x 4- C ..x + xx 4-x See. 


Now, the quotient of 71 — tv divided by a; — a: being, by 
the lafi: Article, equal to g x n when x is = at, it follows, that 
g X n X , or its equal g + ) § , Aa: 4- #Ba: z -f“ gC x * & c - 

is = A 4- 2Bx 4 “ 3Ca: x 4 ~ 4^*’ &c* 

From whence, by comparing the homologous terms, A is 

found = g, B = li, C = ii, D = &c. 

2 3 4 


Therefore n x is = 1 4- g x + g — 4 - — 4 * —— &c. 

2 1 2.3 ‘ 2.3.4 


From the equation n X = v, it is now propofed to fnd x in terms 
of n and v* 


If 


s° 


THE RESIDUAL 


If a; be aflumed — mxv — i -f- A . — i] -j- B . v — 1 1 5 &c. 
and ri' = v, .(m. A, B, £?r. being fuppofed independent of v y ) 

we fhall have -—-;--- 

x —mxv — i -j- A. v — 11 + B.v — i \* &c. 

and, by fubtra&ion, 


X — x — mXv — w + A • V— i| — ^T— i| 1 -j- B . v — j[ 5 — v _ 1 1 3 iSc. 

From which lafl equation, by dividing one fide by n — n* 
and the other by v — v [— v — i — v — i), we get 

X - X - - ■ ■ _____ _ 

” 7 “—“ = m X i -f- A . v—i -f-v— i -f- B . v - 1 1 v-ix v-i 4- ^ H 1 t$v. 
n —n‘ ' ' > 

Hence, by taking x equal to x , and v equal to v 3 we have 


- i -;=»/XI+2A.D-I -f 3B.®— lf-f- 4 C.D— l|‘ (Sc. 

gx n 

g x n , by Art. 2, being the value of the quotient of n — n> 
divided by x — when x is therein taken equal to x. 

But — is = L = —— r =—, which, by divifion, is found 
n v 1 + v —1 

equal to 1 — r o —- 1 -|- v — 1 1 — <u — jj 1 1 —. 

It is evident therefore, that 

J - ^ 1 J _ 4- 

-Xi — V -I ~\- V -1 1 - V — i| -f- v — i] — (sc. 


is = /»xi + 2A.'V-i + 3B.vri] 4_ 4 c.^T7] +5D.^m| 4 & c . 

From whence, by comparing the homologous terms, we find 

m = —, A = — —, B = JL C = — —, D = —, &c. and 

g 2 3 4 5 


confequently a: = — x v —1 — - — i 4 - -— v ^ 
J g 2 1 3 


-\-(£t. 

5. By 



Chap. III. 


ANALYSIS. 

S' 

By putting i + u equal to n = v, we have, by Article 3, 

i + « = i+** + ££+ + j£L @ f . 

1 1 2 2.3. *2.3.4 


and from the equation 



Corollary I. Logarithms being an artificial Set of numbers 
correfponding to another Set of numbers in fuch a manner, 
that the fum of the logarithms of any two numbers is equal 
to the logarithm of the product of thofe two numbers * and 
the fum of the exponents of any two powers of any given quan¬ 
tity being equal to the exponent of that power (of the fame given 
quantity) which is produced by multiplying thofe two powers 
together; it is obvious that the exponents of the powers of any 
given quantity are logarithms of the values of thofe powers 
relpedively : For inftance, a , b, c , d , &c. are logarithms of 

n , n , n, n , &c. refpeftively. 

It follows therefore, from what is faid above, that 


-!L=A. — &c. -Ix«-i+iL 3 -Gfr. 

_3_ g 2 1 3 

an ^ ~3 — afe lo S arithms v > 1 + u > 


* From this property of logarithms, it plainly follows, that, b , r, p, and N 
being any numbers whatever, the Log. of b — the Log. of c is = the Log. 

of —* and p X Log. of N = Log. of N^, 


and 



32 


h 


the residual 


and I + iv refpcaively, in the fame Set, g being of any deter- 
minate value whatever. 6 / r 


From the equation. Log. of i 4 u — -x ~u — & c . 

we, by writing ± in dead of u, have Log. of 7+1 (= Log. of 

i+u-Log.ofu) = jxu-‘ — OZ: + JZl_ &C ' an d 
confequently 


Log. of i -j- u = Log., of u -f- 1 x — iL_I -4- 1 LJ _ 

g 2*3 

which lafl: ferie s con verges when the feries in the former value 
of the Log. of i -(- u does not converge. 

So from theequat. Log. of*u = -x v—i _4.EEH - & c 

g 2*3 

(where u muft be a pofitive number lefs than 2,) we, by fub- 
ftituting L inflead of v, find 

Log. of v = -L x 4- — . 1 El 4- 1. EEJI & c 

g V 1 2 V 1 3 S' I 

where v may be any pofitive number greater than -i. 

Now, as in thefe logarithmic expreflions the value of — may 

be taken at pleafure, fo that in any one Set it be always the 
fame j it is plain, that, to any Set of numbers, we may adapt 

as many Sets of logarithms as we pleafe, by taking -L o£ diffe¬ 
rent values in different Sets j and the logarithm of any number, 
in any one Set of logarithms, being to the logarithm of the fame 
number, in any other Set of logarithms, as the value of the fador 

— in the former Set is to the value of -i- in the latter, that fador 
is called the Modulus of the Set. 


Scholium. Z^t/Natier, the Inventor of Logarithms, firft 
gave a Set, whereof the modulus is Unity :—In which Set the 

logarithm 


33 


Chap. III. ANALYSIS. 

logarithm of io is 2.302585 * Of this Set are the logarithms 
now called Napier s logarithms -f*. And he afterwards, with the 
afliftance of Mr. Briggs, (then Savilian Profeflor of Geometry, 
at Oxford,) undertook to compute a fecond Set of logarithms, 
wherein the logarithm of io is 1; (which Set he found would 
be more convenient for trigonometrical calculations than his 
former;) but, Lord Napier dying before they had finished 
their defign, the work was completed by Mr. Briggs, and 
the logarithms of this fecond Set (which are the common 
tabular logarithms) are called Briggs's logarithms. The mo¬ 
dulus of this fecond Set is — ^ = 0.4342944 : for, by 
what is faid above, the logarithm of 10, which in this Set is 
1, is = — x 2.-102585; and, confequently, - = —--. 

g g 2.302585 

Corollary II. From what is faid above, it plainly appears, 
that if, in any Set of logarithms whofe modulus is x be the 

g 

logarithm of any number whatfoever, that number will be 
equal to 

1 +^ + l!f! + €!l 3 + 

2 1 2 *3 1 2.3.4 

Corollary III. By taking, in the laft Corollary, a* equal 
to ~, it appears that, in any Set of logarithms, the modulus is 

always the logarithm of 1 - 4 — 1 — 4 —-—L _l 1 

1 2 1 2.3 2.3.4 * 

(= 2.7182818) ; or, which is the fame thing, of the ratio of 
the fum of that Series to Unity : which ratio is therefore, by 
Mr. Cotes, called the modular ratio. 

And, — x being the logarithm of the reciprocal of any num¬ 
ber whofe logarithm is *; it appears by taking, in the fame Co¬ 
rollary, x equal to — j-, that the modular ratio is that of Unity 

to the fum of the feries 1 — 1 -j- _L «— T 1 _ 

2 2 . 7 ' 2.7.4. 

(= °- 3 ^ 7 ^ 794 )» _ 

* See Scholium to Article 6. f Thefe are alfo called Hyperbolic 

logarithms > for a reafon that will appear in a fubfequent Chapter. 

L Corol- 



34 


k 


THE RESIDUAL 

Corollary IV. It is evident then, that, in Napier 1 s Set; 
(whereof the modulus is i,) 2.7182818 is the number whofe 
logarithm is 1 ; and that 0.3678794 is the number whofe 
logarithm is — 1 : Alfo, that, n being = 2.7182.81 

M=si + * + !* + 

2* 2.3' 2.3.4 

6 . 


Napiers Log. of 1 + x being, by Art. 5, = x —• - 4* - — 

— + Gfc. it appears, by, writing —* in (lead, of x, that Napier's 

Log. of 1 — x is = — x —> --— —- — and, by fub- 

2 3 4 _ 

traftion, that Log. of i + x — Nap. Log. of 1 — x (= 
Nap. Log. of y~~) 1S e q ua l to 2 xx-|-y 4 -j + y Gfc. 

By this theorem the logarithms of fmall numbers may be 
eafily computed. 

Example I. To find Napier's logarithm of 2. 
Suppoling ii? = 2, x will from thence be found = —, and 


therefore Nap , Log. of 2 is 
= °.69314718. 


+ — + — 
3 • 3 1 '5-3 




Example II. To find Napier s logarithm of 
From die equation = ~ we find x = -^ j and, con- 

fequently, Nap. Log. of i = 2 x -L- + -i- s &c. 

= 0.2231435.5. 

Scholium. To find the logarithm of a great number, by the/ 
fame theorem, we muft find the logarithms of fuch fmall numbers 
as, being multiplied together, fhall produce that great number; and 
then, the fum of the logarithms of any numbers being equal to 
the logarithm of the product of thofe numbers, we fhall, by adding 
thyfe logarithms together, find the logarithm required. 

Example* 


35 


Chap. ID. ANALYSIS. 

Example. To find Napier's logarithm of io. 

It is obvious, that 10 is = 8x^ = 2 x 2X2X-; there- 

4 4 

fore, by what is juft now faid, Napier's logarithm of 2 being 
computed, and likewife his logarithm of the fum of this laft 

logarithm ^dded to three times the former will be (=2.30258509) 
the required logarithm of 10 . 


7 - 

Other theorems, more convenient for computing the loga¬ 
rithms of tome certain numbers, may he deduced from the 
above ; of which the following are examples. 


Example I. To find the logarithm of n, having the logarithm 
°f n -*■ I and n -f-1 given. 

Suppofing or its equal » we from 

n — ix»+i ^ » —* i-r 

thence find x = -—. Therefore, by the theorem in the 
laft Article, Nap. Log. of - r l-- (= 2 xNap. Log. of n — Nap . 
Log. of n — 1 — Nap. Log. of « + i) is = 




3 .2»* — if 


+ 


-&C. 


s •— q 

and, confequently, 

Nap. Log. of n — Nap ' L ° s ' - + Nap ' tk of . 




+ 


V + 


3.2* 1 — 11 5.2« l — l] 


■&C. 


This theorem is very convenient for computing the loga¬ 
rithm of any large prime number. 


Example II. To find the logarithm of a large number n, 
having the logarithms of n — x and n -)- x given. 

F % The 



36 


THE RESIDUAL 


* 


The Log. of i — x being = — Mod. x x — -f- -j &c* 

the Log. of -^zr x is = Mod. x a: -f- ~ + y which (the Log. 

of being = 2 x Mod.x* -f- — + — $0 is = ~ x the 
* 3 5 * 

_ v X* xl ^ _ 

Log. of ~ ~ ■■■ * ■ = 4 x Log. of l -±I * 

I — * , * X s 2 ° T - x 

x -4--C ffc, 

__ 3 S 


I -f- + — + nr Hence, by fubftituting — inftead of 

2 12 1 oO 0 /j 

x, we have Log. of —£— (= Log. of w — Log. of « — x) = 


T * L °g- of ~^ l + Tn + + 18 ^ ^ Therefore, 

the Log. of being = Log. of n * — Log. of n -^~x, the 
Log. of n is =± x Log. of n -f- at + L x Log. of n — * + 

Log. of — Log. of n — x T~~ y 7 * i ? ~ 

- 2 - X S + 7^ + 78^ ® c \ 

This laft theorem is very ufeful in enlarging a table of 
logarithms. 


8 . 

It may lometimes be of ufe to obferve, that, when v is a 
very large number, the Log. of 1 -f- -L will be = ~ nearly, the 

value of the feries -- 1 - -L-l — gfr. (See Art. 5. 

Cor. 1.) being then fo very fmall that it may be neglected. 

_ u 

Therefore, v being as juft now mentioned, if i + Ljbe = N, 
the Log. of N (— u x Log. of 1 -f- will be nearly equal to—. 

9. By 



Chap. III. 


37 


ANALYSIS. 

9 - '-I 

By Art. 2. [x »*] is = gn*, ^teing (by that Article, and 
Cor. 1. Art. 5.) = Nap. Log. of n. 

But in Napier's, Set, 2.7182818 is the number whofe logarithm 
is 1, as appears by Cor. 4. Art. 5, 

Therefore if n be = 2.7182818 (/. e. if the ratio of n to 1 
be the modular ratio) g will here be = 1 ; and, in that cafe, 

[x-l n] is = n ; or, v being = n y [x ^ «u] is = v. 
Confequently [x.'-'u] x [u^.x] being = i, (by Chap. 2. Art.9.) 
[ v -s_ x] is = -L, where x is Nap. Log. of v. 

Moreover, by Chap. 2. Art. 8. [*1; _l x] is equal to —" = —^ : 

[« — V 1 

this laft expreflion mull therefore be = -L • and, confequently, 

[«^x]=pL.±j2, 

u being any fundion of v, or x. 

& For brevity fake, we fhall, in future, put L : x, L : a bx , 

L : x -I*- s /1 -{- x 1 ^, &c. to denote Nap. Log. of x, Nap. Log. 
of a bx. Nap. Log. of x -J- s/ 1 -J- x z \ , &c. relpedively. 
Example I. Suppofe v = 1 -J- z : then we have 

Example II. Sitppojing v = a -j- bz*, we have 
[u L : a-\-bz m ] — a + __ ~~ T Q — <1 ' 

* + a -f b% m 

Example III. Let v be fuppcfed = a -\- z s/2az-f-z\ 

Then we have [u +.L : a z -\- s/^az^fz 1 ] = - J" - z l 

•/ 2 az -}- z x% 

[u 




3 « 


k 


THE RESIDUAL 


[u a -f -z + >/zaz + z‘] being = x[uz. »]. 


Example IV. [a jlL : z n + vV + z 2 *] is = 


'V + z" 


[a z n -f- V'a* + z 2 "] being =.** -f- vV -f«z" X ” ; 


+ * 


Example V. v being fuppofed ~ we have 

_ T * 4, a-1 2gQ -t- z] a -f* z ^ l u ~ * 3 . 

- L: J = -=r r±-s = 

Example VI. Taking v = g ~*>VrA » we find 
° <2 + W -f- z 1 

r„ , I . <■—v~-Tg -| 

^ a -J- v^tj 1 -f- z 1 J zVa* -f- z* 


IO. 

By the laft Article we have [x — L : r w ] = l *~y Now 

L: r v being = vxL:r, [x ^ L : r'] is = [x ^ y xL : r] = 
[x j-dxL: r, v being any function of the variable quantity 

L r , q 

*, and r invariable: Therefore — --- - - is then = [x ^ *u] x L : r ; 

and, confequently, [* _^r ] = r x [x _i_ v] X L : r. 

But if both r and v be functions of x, [x x L : r] 

(= [x L : r v ] = ~^ 1 ) will be = [x *■•] x L : r + 
v x £l. ~ : Confequently [x r^J will then be = r x [x J- i>] 

xL:r A-r~\[x^.r]. 

By proceeding much in the fame manner, may the values 

of [x ^ 71 ”] and [x j- rl^J, &c. be found in terms of r, v, 
w, x, [x-i.r], [x , [x j. w], &C- Exam- 


39 


Chap. III. ANALYSTS. 

Example I. N and n being invariable, [x ^ N”*] is 
= n N”'xL:N. 

Example II. If a, b t and n be invariable ; [x a 
will be = n x a + bx( x x L : a -f- bx bnx x * + bx\ n *~~ \ 

Example III. If (A, N, and a , &c. being invariable) 

AN* be == y-\-a\y-\-b.y-\- c (;r). 
we, by refidual divifion, fhall have 

AN*x[y_i_*]xL:N = y + £.y-}-c.y-|-^ — i) 

+ y + a . y+-c . y-\-d (n -1) -f- .>’+<*•/+^ . y+</ 


I I. 

By the binomial theorem, inveftigated in Article i, we have 

= i. + „* + . «' & c . 

2 1 2t . 3 

from whence, by refidual divifion, (*u —v being the divifor, and 
the value of z independent of v,) we g.et 

i + z| kL:i = z -f v _j_ v , v — ai^L&c. 

J v.V -1 J 

Hence, fuppofing v =- o, or any pofitive integer, we find 
L:i + 3: P 


1 I ■ I • l • 3 (v + i) 

r+^r ' 1 ‘ T+Z 1- 


1 • 1 • 3 (v + 0 


■v +• » 


" + 3 


v + 4 


*-3-4j(v+i). 3-4-5(* + 0 4-5 •&(* + !) ^ 

P being put for z11 + v.»— 2 j^l 

v.r— i J 2,3 


(*)• 


12. Again, 



40 


THE RESIDUAL 
12. 


V-J-I A'4-l 


Again, by the binomial theorem we have z~^~a\ =. 

- x -|- i • d* 

Therefore 


+ *+7. dz* + i±-L iId' z x - , + *=± d'z*-* (sc. 

11 1 2 • 2*3 


71 * + * *+» 


a* is = ^ *- -dz *~ 1 _ V +~ 2 tfc. 

d. x I 2 2 * 3 

Hence, expounding z by a, a d, a - f- id, &c. fucceflively, 
we have 

* — ±±JL ~ e ■ _ *d. a — 

i. x + 1 2 

a ~ 1 &c. 

2 • 3 


« + 2 ^| — a -f- a | 


• a + 4 — 


’ + 4 — r- 7.7+1 2 " 

*—r-V 1 . 2 

2-3 


a \ id\ = 


d . x 4 - i 


±JL=^d\T+ 7 d\ &C. 

2.3 

(sV. 

Confequently, we get, by addition 


$V. 


S = 


a -f- nd] — 


+ 1 * + 1 


- 


x - x -'d t S x ~ 2 &c. 


d.x+l 2 T- 2 • 3 - 

S* being wrote for a 4 - “ -f- “'I '' 4" a 4 - 2 ( 4 * 4" a + 3^ M> 
5*- 1 for /-+ 7+1"-'+ 7+71"-+ 2+W‘H 


From 


41 


Chap. III. ANALYSIS. 

From whence it follows, that 


5 *-'i s== £± 3 _^ 

— dx 


- 2 __ a+nd\ X ' t —a *' 1 _ * — 

t d. x — i 2 

&C. &C. 



*- r • x ~ 7 d 1 S x ~^&c 


2 * 3 - 

-VS*-3. 

*- 2 ' x ~3 j* -4 


2-3 - 

Gfc. 


s*- 2 , 

£* ~~ 3 , GV. being fub- 


flituted fuccefiively, for their refpedive equals, in the value of 
S*, we find 




■ 2^1 + * + 3*1 (») = 


1 \ 7^n3\* + l I f a 4- nd\ x 

i~- FTl_/ + - 


Ad\ a 4- nd[ 1 . 

2 1 * — i ' 

L — a 

. x-i .x-i.Ad 3 ] a + nd l*~ 3 , x.x-i .x-2. x-2. x-4- . A> f + W ]*" 5 
2 -3-4 1—^-3 1“ a. 3 . 4 .5-6 


A. being = = +, 


2 * 3 


= 2T5— ~A=-L, 


30 


A 

= -i- Aa— 

2.7 2 

6s4 a 

2.3.4 

_ r 

42 * 


A 

— 7 . A. - 

i^A 

8.7.6.5.4 V'_ 

1 


2.9 2 

2.3.4 

2 - 3 * 4 * 5 * 6 

3 °* 

V 

A 

— 9 i£A _ 

10.9.8 V 

10.9.8.7.6^ _ 

10.9.8.7.6.5.41" 


2.11 2 

* -.A 1 

2 ->4 

/ 

2.34.5.6 

2.34.5.6.7.8 



# 


[=& 



<&T. 


&c. 


Scholium. When x is = — 1, the numerator and denomi- 

nator of the fraction «-+ + '■~ £_ + * /_ a+nd\‘ +, \ 

d.x+i v rf.*-+q_/ + I 


G 


both 




42 


k 


THE RESIDUAL 

both vanilh.—In that cafe -ixL: - — is the value of that 

, da 

fradion, as appears by what is already taught. 




Writing 2 n inftead of », we have, by the preceding article, 

a — a -\-dy At Ar a-\-\d[[ 2 n) — — l — \ aJ r 2n fi 

*•*+<[_/ + I 

_i_J a + 2 nd\* f xkd\a-\-md\~ l t x .x-\.x- 2 kd'\ a-\-2nct[-^ 
n-a + *1-*'— +. 373.4 [_/-3 


and, writing 2 d inftead of d> we have, by the fame article, 
a + * + 2 d\ + a + ^ v -f a + 6 J[ x (*) = —== | f 

?r 3 

“3 


_J[jrf + 2wd|* ( + * 2^.x-I.*-2.A</ 3 [*+2/fc/f 

*1-/ + 2 l-a*- 1 + ~-3.4 l_/-3 


G?C. 


x+i 


It appears, therefore, by fubtradtion, that 

Z+A X + i+ 33 T + * + PC (*) is = — + /f|' 

~f- 2 »d/|* 1 2 3 — l.x.x-i.x.-2.Ad 3 \a ind)[ 3 _ 

2 1—/" 1 “ — w l -/- 3 


2*- \,X .X-— l.X - 2 .x- 3 . AT- 4 . A</ S 

2-3-4-5-6 


f * + 2^ 

L — Cl 


■*— 5 


Gfc. 


Schol. When a: is = — 1, the value of 


2d 


, r^+^r +i 


is = -I x L : i± 2 Hd } by what is done above. 
2 d a J 


14 . Taking 


Chap. III. ANALYSIS. 


Taking the laft equation from that which immediately pre* 
cedes it, we have 


a — — ^+ 3^1 *+ <* + 4 — * + 5 ^f( 2 *) = 

I ^ a -f- 2 W|* t 2 l —I. 1 , 2 *—I.*.■*-!. x-2.Ad* 

H — a + “ l -/- 1 + 2 ‘ 3 * 4 

^ <2-j-2tf*7j 3 , 2 6 —I. jf. Ar-i . *— 2. *- 3 . x- 4 . AV J j* # “f* 2 W] 

[—/“ 3 + 2 *3*4-5.6 l—/" 5 


» being any pofitive integer. 


By refidual divifion, (x— x being the divilbr, and the values 
of a and d independent of x ,) we get, from Art. 12. 
a x L : a -f- a d\* x L : a -|- d + a + 2^* x L : a + 2d 4- 

*~+~$T x L : * + 3 d (n) 

_1 [a x L \a-\-nd _if a + nd\ x L : 

. *A</' a + nd\ 1 x L : a -f- nd 

J| — a~ l x L : a ** 

..,+Tl‘l_/+> + »l— + xx-A 

j£_Vi + «/f“ 3 

2 - 3 ‘ 4 ^—— 3 w<r ' 


Corollary 



THE RESIDUAL 

Corollary I. Suppofing x = o, we find 

L: a L: a -j- d L \ d 2 d L : a 3 d (ti) 
— yXL :a + nd — -j— jxL:u, — n -J- 


J 


j: Ad f a-j-ndj , Ad* \ a -if nd\“ ^ , A</ 5 J + #4 

H-- +3 *l -.-3 *'■ 


Corollary II. Taking, in the preceding Corollary, a , d t 
and n each equal to 1, we have 


X L : 2 = i 


2 3 —I . A . 2 s —I . A 


2*. 3.4 ” 2 s . 5.6 


&C. 


16. 

As the principal theorem in the laft article was deduced from 
Art. 12. fo from Art. 13. we deduce the following theorem, viz. 

a ~\~d\ xL :a-\-d #-[-34*xL:<z-}-3*/-|- a -{- 5 */j*xL \a + $d( n ) 
_1_f a + 2.nd[ 1 x L : a -f- 2nd xkd 

) a -f* 2 nd\ m ~ 1 x L : a + *nd _ 

1 * — 1 T 

i — a x L : a 

Aa -j- 2nct\ x 

T W- X “ 

2 1 —i. Ad* ^ a + 2 nd[ ^ 

2, 3*4 |_/-3 ^ 

Corollary I. Hence, fuppofing x = o, we find 


■1.^21 

x• x- 2 |x 

X.X- I J 


— I f 4- 2#4* 1 


L : a -j- d -f- L \ a ~\~%d -f- L ’• a •4" 5 ^ "i" L : 0 "{"7^ 00 


Chap, III. ANALYSIS, 
i-f »xL:«+2»</— jjXL :a 

_ _ kd\a-\- znr^ 1 _ 2 1 —1. Ai 1 fa 4- 2W</| 3 

=< - l |_r' _ 3 4 l-a - 3 

2 s — i . A^ s f a -4- 2nd\ * 

, —^-{_v *■ 

Corollary II. Taking, in the preceding Corollary, <z = 2 , 
d = 1, and writing n — i inftead of #, we have 
•L:i + L:3 + L:5 + L:7(») = L:iX3XSX7(») 

. A . 2 J -1. A , 2 s —I . A 

T „_r n , , 1 + "27777 + 2 *. 3.4 + ~7T &C - 

= L :2 *n __ 4 __ 

A 2 3 —1. A 2 s —1. A 

— n — —— rrr-7 — „ v ■ -7 &c. 


, A , 2 3 —1. A . 2 s —1. A 

T *-1 , , 1 + THT 7 + 

= L:2’-V+< __ _ __ 

_ _ A 2 3 —r . A 2 5 —1. A 

n 2.1.2 .n 2 3 »3.4.» 3 2 5 .5-6.« s C ' 

= «xL:» -)- «-f-J-xL :2 , — n --- 2 3 ~~* — 

1 1 2 * 2. I . 2 • W 2 3 .3- 4.W* 

2*—i . A rJ 3 r » • , A . 2 3 -—1 . A . 

I xL:2 bein § = 1 + 7 ^T + 1 mT + 


2 ^ , | ^ 

a> ; 6 - GV. by Cor. 2. of the laft Article. 

Other theorems may, in like manner, be derived from Art. 14. 
which we may take notice of in an Appendix to this Treatife; 
and perhaps add fome farther improvements on the fubjed of 
the laft five articles, which fome time ago engaged the attention 
of Mr. De Moivre, Mr. Stirling, and other eminent mathe¬ 
maticians. 


THE 



46 THE RESIDUAL 

I CHO>icw>o>E>>c!oa<>a I 

)it«sW«UW8sW<Wi5&<Wi9esWei^ 

THE 

RESIDUAL ANALYSIS. 


CHAP. IV. 

Of the Properties of certain Algebraic Expressions. 

H E articles in this chapter will be found of con- 
liderable ufe, in geometrical and phyfical enquiries; 
and, to the end that we may proceed with as much 
perlpicuity as poflible, it is thought proper to infert 
them here, previous to fuch enquiries. 

r. 

Suppofe E to be an algebraic exprefpon compcfed of x and other 
quantities ; and Juppofe , tbat> bow near foever x be taken to fome 
certain quantity g, E is pofitive when x is lefs than g, and negative 
when x is greater than g; or pofitive when x is greater than g, 
and negative when x is lefs than g : then Jhall E, or its reciprocal , 
be — o when x is = g. 

For it is well known, that o is the only limit between pofi- 
tive and negative 3 and it is therefore plain, that either the value 
of E muft continually approach to that limit, or increafe or 
decreafe without limit, upon taking x nearer and nearer to g. 
If the value of E approaches to o when x is taken nearer and 

nearer 



47 


Chap. IV. ANALYSIS. 

nearer to g, it is evident that E will be = o when at is = g: 
If the value of E increafes or decreafes without limit when x is 

taken nearer and nearer to g , the value of L will then con¬ 
tinually approach to o; therefore, L being pofitive or negative 
according as E is pofitive or negative, it is evident that, in fuch 
cafe, -L will be = o when * is = g. Confequently either E 
or its reciprocal muft be = o when x is = g. 


2 . 

Q J>eing an algebraic exprefjion fo compofed of x and other quan¬ 
tities, that neither it nor its reciprocal vanijkes when x is therein 
taken equal to g ; the faid exprejjion , when x is greater or lefs than 
g, between certain limits , will (fuppofng it not to become imaginary) 
be pofitive or negative , according as (q) the value of Q, when x is 
equal to g, is pofitive or negative . 

For, let A and B be pofitive quantities ; and fuppofe that either 
Q*J>r ^ is = o when x is = g -f- A, and likewile that Q^r 

~ is = o when x is = g — B, but that neither Qjior is = o 

when x is taken between the limits g -|- A and g — B : then, it 
is obvious, that, whilft x is taken between thole limits, Q^(if 
it does not become imaginary) will be always pofitive, or always 
negative; and, confequently, pofitive or negative according as q 
is pofitive or negative. 


The fame conclufion follows, if, inftead of Q^or L being 

= o when x is = g A, or when x is = g — B, thofe ex- 
preifions then become imaginary, but are real and finite when x 
is taken between the limits above-mentioned. 


3 - 

Let m be an odd number, or a fra&ion whofe numerator and 
denominator are both odd numbers; and P any algebraic ex- 
preflion fo compofed of x and other quantities, that its value 
fhall be real when x is either greater or lefs than g , (between 

certain 



THE RESIDUAL 

certain limits,) and that neither it nor its reciprocal (hall vanilh 
when x is equal to g alio let p be the value of P when x 
is =£- __ 

Then, feeing that aT— g] w is pofitive or negative, according 
as a: is greater or lels than g; it is evident, (from what is (aid in 
the laft Article) that, how near foever x be taken to g, if p be 

pofitive, a; —g|*x P will be negative when x is lefs than g, and 
pofitive when a* is greater than g; or, if p be negative, the fame 

exprefiion (at — g\ m x P) will be negative when x is greater than 
g, and pofitive when x is lefs than g. 

Corollary. Suppofe E to be an algebraic exprefiion com- 
pofed of x and other quantities : Then, if E be pofitive when 
.v is greater than g , and negative when x is lefs than g, how near 
foever a: be taken to g j it is manifeft, from what we juft now 
obferved, that 

x — g\ m x Qjmay be aflumed = E ; 

m being as above (perilled, and Q^fome algebraic exprefiion 
confiding of fuch quantities, that its value (hall be real when x 
is either greater or lefs than g, (between certain limits,) and that 
( q ) its value when x is equal to g (hall be finite and pofitive. 

Moreover, if E be pofitive when x is lefs than g, and ne¬ 
gative when x is greater than g, how near foever x be taken to 
gy it is likewife manifeft, that, in this cafe alfo, 

x — gf* x Qjnay be aflumed = E ; 
m and Q^being as before fpecified, except that q , inftead of 
being pofitive, muft be negative. 

Hence it is evident, that, when x is = g, (whether q be 

pofitive or negative,) E or T will be = o, according as m is 
pofitive or negative^ which agrees with what is faid in Art. i. 


Chap. V. 


ANALYSIS. 


49 




* 


Mud 


B 3 >n« 


$■<&><& 

n , ^'#Ser > 


W*&M 


% 


T H E 

RESIDUAL ANALYSIS. 

CHAP. V. 

Of the Tangents of curve Lines. 

i. 

5 fi( 50 ( 5 @O 5 O 0 ( H E curve AyPy, having its convexity downwards, as Fig. i. 
gj^k.^C j n Fig. i. being referred to the bafe AB *, if the right 
^ ne NrPr touch the faid curve in P j and, b rq being 
) 9 ()eC)e()e 0 eC parallel to BP, if AB be called x ; BP, y j Ab, *; 

by, y i and the fubtangent BN, s : then will bN be = j— xr+x, 

* / 

hr = y — y X ^ — x, and the refidual by — br (= qr) = 
y xx — * — y — y. Now, bry being drawn on either fide 

of BP, by is manifeftly greater than b r ; therefore f x x — a; 

y y va Le of by — hr) muft be always pofitive, when, 

* being of any given value, x is either lefs or greater than *. 

But if the convexity of the curve be upwards, as in Fig. 2. Fig. 2. 
y xx — x — y — y (the value of by — br) muft be always 

H negative. 




5° 


THE RESIDUAL 

negative, when, at being of any given value, * is either lefs or 
greater than x, ' 

Now, fince the expreflion 7 x x — x — y—y mull be al¬ 
ways pofitive or always negative, when, x being of any given 
value, x is either lefs or greater than x , the quotient of that ex¬ 
preflion divided by x — x (viz. dL — [x \y]) will, it is obvious, be 
pofitive when x is lefs than x, and negative when x is greater 
than x ; or pofitive when x is greater than x, and negative when x 
is lefs than x , how near fbever x be taken to x. Therefore, by 
Chap. 4. Art. 1. 

7 — [x jl. y] will be = o * j 

and, confequently, s = —~— . 

O — >] 

Fig. 1. Example I. To draw a tangent to any Parabola , whofe equa - 

m 

tion is ax r = y j m being fuch , that the convexity of the curve 
is downwards. 

We have (according to our fcheme) b q — hr (= qr) = 

— _ m m 

— x x — x — ax r -(- axr- where s mull be of fuch a value, 

that the value of the whole expreflion (hall be pofitive, when, 
x being of any given value, x is either greater or lefs than x. 

mm 

Now if ffLl xx — * — ax r + ax r be always pofitive when x 

m 

— mm 

ax r v .-- — . — 

- X*“*- axr 4. ax r 

is either greater or lefs than x, -i_1_L_, or its 

X - X 

* The other equation (viz.- 1 -- = o) which, if poffible, wou’d 

7 — [* — XI 

follow from the fame article, is, in this cafe, manifeftJy impoffible. 


Equal 


Chap. V. 


ANALYSIS. 


_ ax r 

Equal — 



(found by our divi- 


fion taught Chap. 2. Art. 1.) it is plain will be pofitive when 
x is lefs than x, and negative when x is greater than x. There¬ 


fore, by Chap. 4. Art. 1. the value of this lafl expreflion will be 
= o when x is therein taken equal to x : Which value is 


manifeftly = ~- — . ax r 


5 1 


Conlequently from the equation . ax r 1 = o, the 

fubtangent s is found = -x. 

In this example we have given the procefs at full length, 
that the Reader may the more clearly underftand our doctrine.— 
In future, our examples will, for the moll part, be more concile. 


Example II. To draw a tangent to the Circle AyP^, wbofe Fig. 3. 
equation is 2 ax — x z = y* j the radius AC being = a t AB = x y 
BP = y. 

From the equation of the curve, we have, by relidual divifion, 
a — x =y x [x j~y] j from whence we get [x _l y~\ = —y~ • 

Therefore —~—, the value of s or NB, is = —= ———• 

a ~ x a ~ x 

If BC be called x, and BP as before^ the equation of the 
curve will be a 1 — x z =y z $ and the fubtangent NB will be 

X 

Example III. To draw a tangent to the Rllipfis APD, whofe Fig. 4. 
equation is b z x 2 ax —■ x z = ay z ; the femi-tranfverfe axis AC 
being = a, the femi-conjugate CD = b t AB = x, BP =y. 

H 2 


From 



52 


THE RESIDUAL 

From the equation of the curve, we get, by refidual divifion, 

ab ‘— b'x = ay [x s.y \: hence [x j-y] is found = b — x 

d y 

Therefore —-—, the value of the fubtangent NB, is = 
[*-y] 

a l y L 2ax — x 7, 

b z xa — * « —* 

If BC be called x , and BP as before; the equation of the 
curve will be F x a —** — ay 1 ; and NB will be = a ~* ■. 

If, P d being parallel to AC, Cd be called * ; and dP 9 y : the 
equation of the curve will be a x F — x z = Fy x ; and the 
fubtangent dn = — . 

Fig. 5. Example IV. Let it be propofed to draw a tangent to the 

Hyperbola AP, whofe equation is b 1 x 2ax -|-x == a x y x ; the fetm- 
tranverfe axis AC being = a, the femi-conjugate CD == b, 
AB = Xj BP = y. 

From the equation of the curve we have ab x -\~Fx = a x y [x^y] : 
hence we find \x yl = — x Therefore —, the 

L * > C *±y]. 

value of the fubtangent NB, is = — —-— = 

Pxa + x «+* 

If BC be called x , and BP as before; the equation of the 
curve will be F x x'' — a x = a 7 y x 5 and NB will be = * ■ * . 

If, P d being parallel to AC, Cd be called x ; and dP , y; 
the equation of the curve will be a 1 x F -J- x 1 = Fy* ; and 
the fubtangent dn = . 

Fig. 6. Suppofe CE, Ce to be the afymptotes of the two branches 
AP, AQ, of the Hyperbola PAQj then, BP being parallel to 
Ce } if CB, BP be called x and y refpedtively; the equation of 

the curve will be xy~p\ p being put for Therefore, 

y + 




53 


Chap. V. ANALYSIS. 

>> + x[x.£-/| being = o, [x^.y] is = — j-, and —^—(= NB) 

I *—y J 

= — x ; which being negative, indicates, that N is on the 
contrary fide of BP, from C. 


Example V. To draw a tangent to the Cijfoid AP, whofe Fig. 7. 
equation is ay 1 — xy 1 = x\ 

From the faid equation we have 2 ay [x -i~y] — y z — 2 xy [x _^y] 


= lx 1 i and from hence we get [x yl = 2 *—i . L : Which 
laft exprefiion being fubftituted for [x ^ y] in the equation 
s — —-—, we find s — 

[x-t-yf 3 * +y 


Example VI. To draw a tangent to the exponential curve , 
whofe equation is a =y. 

By refidual divifion (fee Chap. 3. Art. 9.) we get a* x L : a 

= [x -±.y] : Therefore s (= —) is == , an invariable 

l* — y ]/ L : a 

quantity. 

Example VII. To draw a tangent to the exponential curve, 
whofe equation is x* = y. 

From the faid equation we find, by refidual divifion, /xL:x 

-p x* — [x vl: Confequently s (~ -^— J is =---. 

I* — y]/ 1 + L : x 

After the fame manner may we draw tangents to any other 
geometrical, or exponential curve, referred to an axis: but to 
perform the like by our Analyfis, when the curve is a tranfcen- 
dental one, or a fpiral, it will be neceflary to underftand what 
will be explained in the fublequent part of this chapter. 


2 . 

A line has its concavity turned one way, when the right lines 
that join any two of its points either fall upon the line itfelf, or 
on one fide of it, none falling on the oppofite fide.—Here we 
comprehend, not only curves, but likewife fuch lines as have 
re< 5 tilineal parts. 3. When 



5+ 


THE RESIDUAL 


k 


3 • 

When two lines, having their concavity turned one and the 
fame way, have the fame terms, and one wholly includes the 
other, the perimeter of that which includes is greater than the 
perimeter of that which is included *, 

4 * 

Fig. 8. When a curve hyP is convex towards the bafe, and the angle 
BPN, made by the ordinate BP and tangent PN, is acute; the 
ordinate by being drawn, interfering NP in r, P r will be greater 
or lefs than the curve Py, according as Ab is lefs or greater than 
AB. For, drawing a tangent to the point y, (between h and 
P,) interfering Pr in v j Puy, by the preceding Article, will be 
greater than the curve Py. But ru, which fubtends an obtufe 
angle in the triangle qrv , is greater than qv , which fubtends 
an acute angle in the fame triangle : therefore P vr (i. e. Pr) 
will be greater than P^; and confequently Pr ftill greater than 
the curve P q. Moreover, q being on the other fide of P j Pr, 
which fubtends an acute angle in the triangle Pyr, is lefs than 
the chord Py, which fubtends an obtufe angle in the fame tri¬ 
angle : therefore, the chord being lefs than the arc it fubtends, 
Pr will be ftill lefs than the curve P q. 

Fig. 9. When the curve hyP is concave towards the bafe, and the 
angle BPN is acute j the ordinate by being drawn, interfering 
(as before) the tangent NP in r, it is evident, from what is faid 
above, that Pr will be lefs or greater than the curve Py, accord¬ 
ing as Ab is lefs or greater than AB. 


5 - 

fig. 8. Let AB be called x ; BP, y Ab, by, (parallel to BP,)^; 
9 * 

the fubtangent BN, s the tangent PN, t; and the parts hP, hq 
of the curve, z and z refpedively : then will Pr be = — x * —x 


or x x — Xy and the arc Py = z — z or z — z y according as x 


The laft two articles are from Archimedes dt fphara et cylindro. 

is 



55 


Chap. V. ANALYSIS. 

is lefs or greater than x. Therefore, by the laft Article, if the 

convexity of the curve be downwards, as in Fig. 8. — x x — ~ic 

S / 

will be greater than z — z , when x is lefs than x ; and - x x — x 
will be lefs than z — z, when x is greater than x. Hence it 
evidently follows, that, the curve being convex towards the bafe, 
y x x — x will be always greater than z — z, when x is taken 
either greater or lefs than x : confequently the expreffion 
y x x — * — z — z will be always politive when x is fo taken. 

Moreover, if the convexity of the curve be upwards, as in 
Fig* 9. y x x — x being lefs than z — z when x is lefs than x , 

and y x x — x greater than z — z when x is greater than x, 
y X x — x will be always lels than z — z when x is taken 
either greater or lefs than x : confequently, in this cafe, the 
expreffion y x x — x — z — z will be always negative when x 
is fo taken. 

Now, fince the expreffion y x x — x — 2; — 2; mull be always 
politive or always negative, when, x being of any given value, x is 

either lefs or greater than x ; the quotient of that expreffion 

divided by * — x (viz. - [x | z]) will, it is evident, be pofi- 

tive when x is lefs than x 3 and negative when x is greater than x; 
or politive when x is greater than *, and negative when * is lefs 
than x, how near foever x be taken to x. Therefore, by 
Chap. 4. Art. 1. t 


56 


TE-IE RESIDUAL 

— — [.v _£_ z] will be = o * : 
and, confequently, [x ^ z] = ~ = s/1 + L x —/I * > 

j being — —A—, and t — HlZZl L, by what is faid in 
Article i. 

The fame conclulion, it is obvious, will likewife hold true, 
when the ordinates decreafe from h towards P. 

Corollary I. Hence it is manifefl, that the fubtangent, 
tangent, and ordinate, are to each other as i, [x _r_ z ] 9 and 
[x y] refpedtively. 

Corollary II. Suppofe hP to be an arc of a circle: then, 
if x be the verfed fine, y the line, and a the radius thereof; it 

is well known, the fubtangent will be to the tangent, as y to a ; 

and therefore, by the preceding corollary, y : a : : i : [*-1-2;]. 

Confequently [x z ] is then = 1 =z ^ a -r . 

Moreover, [* -i. 2;] being = —-7 (by Chap. 2. Art. 8.), 

ly J 

tL± is = t, and [yjLz] = But, y being = 

[y -*-»] _ } y a _ x 

,/2 ax — .v\ we, by refidual divifion, have i = —====x [y— x ~\\ 

V2 ax-x x 

and, from hence, [y ^-x] — There * 

fore [y+z] is = 

Let u be the cofine of z; then will a — <y* be = u and, 
confequently, [u ^y ] == Now [y z] being = ~ 

* The other equation (viz.---— = °) which, if poflible, would 

*7 — [*-^-*0 

follow from the [fame article, is, in this cafe, impolEble. 


this 



57 


Chap. V. ANALYSIS, 
this Iaft quantity will be = ——-— ; and, therefore, [u _ z\ 

va' — j* 

will be = — _ • = _^f = . 

Corollary III. being = j* -|-y*, we, by refidual divi- 
fion, have t [x t] = j [x j] y [at .<_y]: Hence j x [a: j_ /} 

= [^ I l + 7 x [*-.?]• 

Now AN being called r, j will be = a; — r; and confe- p; g> g, 
quently [a; _i. s] = i — [a; _£_ r] : moreover, by what is faid 

above, -i is = [x x], and = [x Therefore [a:-i_2] x 

O ^.t] is = i — [x r] -f [* = [* j. z ] 1 — [a: r] 

(i + [x j-y ] 1 being = [x -i-zY) ; from whence it appears, 

that [x _l /] is = [x z] — —~ r \ 

i* — Z J 

6 . 

By means of the conclufions deduced in the preceding article, 
we are now enabled to apply the theorem we inveftigated in 
Art. i. to tranfcendental curves referred to an axis. 

Example I. To draw a tangent to the Cycloid AP; whofe Fig. 10. 
nature is fuch , that , the femicircle A/>D being defcribed , and the 
ordinate B/>P being drawn perpendicular to the diameter AD, BP is 
= Bp -|- (Arc) A p. 

Eet AD be called 2a; AB, x\ BP, yBp, u ; and the arc 
A/>, w. Then, y being = u -j- w, [x j.y ] will be = [x_^_w] -j- 

\x _i. w] == [x j-u] - j- -f. But, by the nature of the circle, 
u * s == 2<3 x — x ; from whence \x ^.u\ is found = --- • 

Therefore, by fubftitution, it appears that [*_^y] is = 2 - • 

Confequently —-—, the value of the fubtangent NB, is = 

[x -s-jr] 

—- * * * 


I 


It 


5 8 THE RESIDUAL 

It is obfervable, that 2 ? (= NB) is to y (= BP), as x (= AB) 

to u (= B/>): Therefore the tangent NP is parallel to the 
chord A p. 


Fig. ii. Example II. To draw a tangent to the Quadratrix AFD. 

DE being one fourth of the periphery of a circle defcribed 
about the center C, draw the radius CPG, which fuppofe equal 
to a: call AC, b-, AB, x ; BP, y ; EF, v ; FG, (parallel to 
BP,) and the arc EG, w. Then, by the nature of the qua- 

dratrix, y will be = — ; from whence we have [x = 

— x [x jl w\ = — x [v jl w] x [x_l v] = ilLzzll, ~ - -] be- 

a L J a L J L J u [x -i- v J 

ing = [*u -i- w] by Chap. 2. Art. 8. and [*u w] = by the 

laft article. Moreover, by the nature of the circle, u is = 2 av 

— v 4 : from whence, by refidual divifion, we get u [x _l u] 

— a — v x [x jl *u] ; and hence ~ E follows 

therefore, that [x^.y] is = — ^ > an< * confe- 

quently that [x_j_ *u] is = LtfLziiJ, and [x_i_a] = —j- x [*— f\* 

Again, by fimilar triangles, b — x : y 1: a — via-, therefore 

ay _ vy is = bu — ux. Hence, by refidual divifion, we have 

a[xs-y\ — v [x-£-y] — y [x^v] — b [x^ n) x [x-lu] u 

and confequently, by fubftitution, a — v X [x _)’] : 


j, — X x a x [x _L .y] — u. From whence it appears, that 

x [x juy] is = xX - a ~ - x [x -i-y] — u ; therefore [x ^y] is 
=- hu __ Confequently —the value of the fub- 

uy—xXa — v [xJ -y\ 

tangent BN, is = 4 - ~ 


7. AjPy 



Chap. V. 


ANALYSIS. 


59 


7 - 

A q?q being a curve of the fpiral kind, whofe ordinates Cy, Fig. 12. 
CP, C q, all iflue from the point C; let the circular arc defe be 
defcribed, whofe radius is 1 ; and draw any right line C d, inter¬ 
fering the faid arc in d. Then, fuppofing the right line NrPr 
to touch the curve in P 5 and fuppofing C qr to interfeft the faid 
tangent in r, and the circular arc in e: if CP be called y ; Cy, 
y ; the fine of the angle dCf, (to the radius i,)«; the fine of dCe y 

u ; and the fine and cofine of CPN, s and c refpeftively: the 

fine of eCf will be denoted by uV 1 — u — u*/ 1 —- id or 

us/ 1 — u — us/ 1 — u\ according as de is lels or greater 

than df ; and the fine of CrN, (or CrP,) by cu </1 — u — 

cus/i — u — suu + ss/i - a'xv^i — u\ which lafi: fine, 

/ / 7 

for brevity fake, we will call k: Moreover Cr will be = ^, and 

sy — ky k 

the refidual Cr — Qq [z=zqr) = — y = —Now, C qr 

being drawn on either fide of CP, Cr is manifeftly greater than 

sy — ky 

Cq: therefore ——-!• (the value of Cr — Cq) mull be always 
pofitive, when, u being of any given value, u is either lefs or 
greater than u . 

If the convexity of the curve be towards C, as in Fig. 13. 

sy—ky 

—-— (the value of Cr — Cq) will be always negative, when, u 
being of any given value, u is either lefs or greater than u. 

sy—ky ' 

Since the exprefiion —~ mull be always pofitive or always ne¬ 
gative, when, u being of any given value, u is either lefs or 
greater than u ; the quotient of that exprefiion divided by u — u 

(viz. 


S y — KJ. 

—=) will, it is obvious, be pofitive when u is lefs than 

l '—r 1 2 


THE RESIDUAL 

u 3 and negative when u is greater than u ; or pofitive when u k 

greater than a, and negative when u is lefs than a, how near fo- 

ever u be taken to u. Therefore, by Chap. 4. Art. 1. the 

value of that quotient, when u is equal to a, (i. e. when y is =y,) 

or the reciprocal of that value, will be = o. But, by the 
method taught in Chap. 2. Corollary to Art. 6. the value of 

sy — ky - 

7 —-— , when u is = a, is found equal to s \u yl- , cy — -f- s. 

k.u — u / ^ tjk?~U X 

Therefore s [a y] — *7== is = o *: Hence - = 1 — x —— . 

If the arc be called w 3 1 will be n= [u^w\, by Art. 5. 

V I—a 1 

It therefore appears, by fubftitution, that 

is = ~ == Tang, of the angle CPN. 

Suppofe CN, perpendicular to the ray CP, to interfedl the 
tangent PN in N : then, radius being to CP, as the tangent 

of CPN to CN, CN will be = . 

Seeing c is = s/ 1 — s\ it is evident that-7== is —: 
b Si— s' 

from whence we have s = y -^- ~ ■ . - . 

' / [us-yf+y'l«^tv] i 

8 . 

When the curve AyP is concave towards C, and the angle 
CPN, made by the ray CP and tangent PN, is acute the ray 
C qr being drawn, interfering the faid tangent in r, P r will be 


The other equation (viz. 


s[«^-y]- 


. = 0) which, if poffible. 


S 1 — * 


would follow from what is faid above, is manifeftly impoflible. 


lefs 



Chap. V. ANALYSIS. 

lefs or greater than the curve Py, according as de is lefs or 
greater than df: And when the curve AyP is convex towards C, 
and the angle CPN is acute, as in Fig. 13. Pr will be greater 
or lefs than the curve Py, according as de is lefs or greater than 
df. This may be eafily proved, by reafoning as in Art. 4. 

Now, retaining the notation in the lalt Art. Pr will be equal to 

L x u/i—u — u /1 —id or — L x u/ 1 — u — u \/1 — u\ 

according as de is lefs or greater than df. Therefore, calling the 
parts AP, Ay, of the curve AyPy, z and z refpedtively; it follows 

that the expreflion z — z — Lx u/ 1 —u — 11 /1 —u mult 

be always politive or always negative, when, u being of any value 
whatever, u is either greater or lefs than u. It is obvious then, 

that the quotient of the faid expreflion, divided by u — u 
z — z 1— u x — us/1 — a 2 \ 

(viz. — 7 X - 'u"— u -'/ be P°^ lt i ve when u 

is lefs than u , and negative when u is greater than u > or politive 
when u is greater than u , and negative when u is lefs than u y 
how near foever u be taken to u. Therefore, by Chap. 4. 
Art. 1. the value of that quotient, when u is equal to (i.e. when 
y is =y, and z = z,) or the reciprocal of that value, will be 
= o. But, by the method taught in Chap. 2. Cor. to Art. 6. 

us /1 — u 1 — us /1 — u 1 

the value of-, when u is = u, is found equal 

U - U / 

to — 4 - s/ v — u == ♦ and k is then equal to s. 

VI — u x vi — u 1 

Therefore \u-l. z] - 7== is o *: Hence [u z] = —;===. 

L J ss/i— u l 1 J ss/i—u 1 - 


* The other equation (viz* 


would follow from what is faid 



above, is impoffible. 


o) which, if poflible. 


Seeing 


61 


6 2 


THE RESIDUAL 

Seeing that ^ 1 — is = \u s- w], by Art. 5. it appears, by 

fubftitution, that \u-l.z\ is = y ' ~-— z=:\/[u±yY^ r y"[uj^w] z ; 


s being 


y[u s-w] 


— yY + y r [*—™] 

From what is faid it is evident, that s is 

[» — J J 
O %] 


r, by the preceding Article. 


and 


confequently c (= s/1 — j 1 ) is 

Moreover, fuppofing CQ^o be perpendicular to the tangent 

PQ j radius will be to y, as — (= r) to CQ ; therefore 

[« -l z ] 

CQwill be = ^ : and radius will be to v, as (= c) 

to PQ ; therefore PQwill be == 

[« — 

Corollary. Hence it appears, that the perpendicular CQ, 
the tangent PQ, and the ray CP, are to each other as y [u j_ w], 
and refpettively. 


By means of the theorems inveftigated above, we are now 
enabled to draw tangents to any fpiral whofe equation is given. 

Fig. 14. Example I. To draw a tangent to the fpiral ^Archimedes j 
whofe nature is fuch , that, any circle AF being defcribed about the 
center C, and any ray Of FP being drawn , the arc AF is to FP 
in a conftant ratio. 

Let CA be —m, Cd = 1, the arc df=zw, CP =y ; and let 
the given ratio of AF to FP be as m to n: Then nrw will be 
= AF, and nw =y — m. From whence we have n [u w] -=. 

[u-L-y] j and confequently, by fubftitution, CN (= ——by 


Art. 7.) is found = dL . 


[u^y] 


Example 


Chap. V. ANALYSIS. 63 

Example II. To draw a tangent to the fpiral CdP j whefe Fig. 15. 
nature is fuch , that the arc CdP is to the ray CP in an inva¬ 
riable ratio. 

Let CdP be = z , CP = y; and let the invariable ratio of z 
to y be that of a to b : Then bz will be = ay. From whence 

we get j [u y] = \u ^ z]> which, by the laft Article, is 

= s/[u -!~yY~\-y~ [ u — w]\ Hence it appears, that [» _j_ y] is 

= ~=L= X y \u -i- i and therefore - — ■ tV \ the tangent of 

v a 1 —- b x [« -£-j] 

the angle CPN, is = . and CN = ‘tZEE xy . 

It is obfervable, that the angle made by the tangent PN with 
the ray CP is invariable ; which is a known property of the loga¬ 
rithmic fpiral. 


I O. 

SuppoJe the moveable curve aB to revolve along the immoveable Fig. 16. 
curve AB, fo that the arcs aB , AB be always equal ; and fuppofe , 
that , during the motion , the point P, having a certain pojition with 
refpett to the curve aB , deferibes the curve OPQ, the curves and 
the deferibing point keeping always in the fame plane : then ,, if 
from B, the povit where the two curves , aB, AB, touch each other 
when the deferibing point is in P, BP be drawn ; PR, perpendicular 
thereto , Jhall touch the curve OPQ Jn P. 

For, about the center B, with the radius BP, delcribe the Fig. 17. 
circular arc EPF : and, having drawn BFQ, fuppofe that, when 
the deferibing point is in Q, the moveable curve is pofited in 
tfbD ; b being that point thereof which was at B when the 
deferibing point was at P, and D being now the point of contact 
of the two curves abD, ABD : join bB, bQj and let ef touch 
the curve <zbD in b, and meet the curve ABD in f —Then, 
becaufe the arcs BD, bD are equal; and bf -j- the arc T)f is 
greater than the arc bD, bf lhall be greater than the arc Bf 
and confequently Fill greater than the chord Bf: Wherefore 
the angle bB/' (made by bB and the chord Bf) will be greater 
than Bbf and Bbr greater than bBg, made by bB and the 

conti- 


64 * ' THE RESIDUAL 

. continuation of the chord fB. It is evident therefore, that the 
angle BbQ, which is = Bb^-|"^Q, will be greater than QBb, 
which is = bBg — QBg •, and confequently BQjviil be greater 
than bQ^ But bQJs manifeflly equal to BP, being the fame line 
transferred with the moveable curve. Therefore BQ^is greater 
than BP, i. e. than BF. Hence it appears, that the point Q^is 
without the circular arc EPF. 

Fig. 18. Suppofe now, that, when the deferibing point is on the other 

fide of P in O, the moveable curve is pofited in adb ; b being 
that point thereof which coincides with B when the deferibing 
point is in P, and d being now the point of contadt of the two 
curves adb, AdB : and join BO, Bb, bO, b d. —Then, the arc 
Bd, which is equal to the arc b d, will be greater than the chord 
bd. But, Be being drawn, touching the curve AdB in B, and 
interfering the chord b d in e, Be -(- ed will be greater than 
the arc Bd ; and therefore Be will be greater than be: Where¬ 
fore the angle Bb^ will be greater than bB*. It is plain then, 
that the angle BbO, which is = Bbe -|- fbO, will be greater than 
bBO, which is = bB<? — eBO ; and confequently BO will be 
greater than bO. But bO is evidently equal to BP, being the 
lame line in a different pofition. Therefore BO is greater than 
BP, i. e. than BE. Hence it appears, that the point O is without 
the circular arc EPF : Therefore, the fame being proved of the 
point Q, it follows, that the faid circular arc touches the curve 
OPQjn P. Confequently PR, which is a tangent to that circular 
arc, will alfo touch the curve OPQ in P. 

Whatever pofition the deferibing point P may have with refpedt 
to the moveable curve, and whether that curve revolves along the 
convexity or concavity of the immoveable one, the tangents to 
the curve deferibed by P will always be determined by the rule 
here given ; and in no cafe will the demonflration differ greatly 
from the above.—The deferibing point may indeed be fo polited 
with refpedt to the moveable curve, that the circular arc EPF 
fhall fometimes fall without the curve OPQ, which occafions 
fome little difference in the demonflration ; but there will not 
then be any particular difficulty in it: Any farther explanation is 
therefore unneceffary. 

This article (which was fuggefted by one in Huygens Horolog. 
Ofcillator.) will be found of great ufe in many enquiries concerning 
cycloidal curves. Scho- 


Chap. V. ANALYSIS. i 

Scholium. If the point P, inftead of keeping a certain po- Fig. 
fition with refpe< 5 t to the curve aB, be fuppofed to move infuch 
a manner along the bafe MN of that curve, whilft the curve 
itfelf revolves as before-mentioned, that any ordinate being drawn 
from the point of contadl B perpendicular to the faid bafe, P 
(hall always be at the end of the correfpondent abfciffa ; then, 
by what is proved above, will that abfciffa (or the bafe) be a 
tangent to the curve OPQ^defcribed by the point P during fuch 
motion. 

I I. 

With refpett to curves referred to an axis, it is obvious, that Fig. r. 
when the convexity of the curve is downwards, as in Fig. i. the 2 - 
value of the quotient of the ordinate divided by the fubtangent 
increafes as the abfcifla (*) is taken greater and greater: and when 
the convexity is upwards, as in Fig. 2. the value of the faid 
quotient decreafes as x is taken greater and greater. Now, by 
what is faid above, that quotient (refuming the notation in Art. 1.) 
is equal to [x j-y\. Therefore, in the former cafe, [*-!-/) — 

[x^.y] will be pofitive when x is lefs than x, and negative when 

x is greater than x : and, in the latter cafe, [x ^y] —- [x-t-y] 

will be pofitive when * is greater than x , and negative when x 

is lefs than x . It is evident then, that, x being either lefs or 
f *—y] — t*—y] 

greater than x , -be a i wa 7 s pofitive or always 

negative, according as the convexity is downwards or upwards. 

Hence it is plain, that the value of [x^y] (fuppofing both it 
and its reciprocal to be finite) will be pofitive or negative, accord¬ 
ing as the convexity of the curve is downwards or upwards. 

i a. 

With regard to curves referred to a fixed point C, it is evident, Fig. 12. 
that, when the concavity of the curve is towards C, as in Fig. 12. 13* 

the perpendicular from C on the tangent increafes as the ray or 
ordinate (y) is taken greater and greater: and when the con¬ 
vexity is towards C, as in Fig. 13. the faid perpendicular decreafes 

K as 


Cm 


66 


THE RESIDUAL 

as y is taken greater and greater. But, by what is faid above, 
that perpendicular (reluming the notauon in Art. 7. and 8.) is 

equal to - -. Confequently, in the former cafe, 

[« - *] (# — zj 

y 1 [u — us] 

-will be pofitive when u is lefs than u , and negative 

[u z] 

when u is greater than u: and, in the latter cafe, 

' [u -L. x J 

y* [u — w] 

- - - will be pofitive when u is greater than and 

negative when u is lefs than u. Therefore it is manifeft, that, u 


y x [u -i- us] y 1 [u -L. Us] 

being either lefs or greater than u , --—i-L -h 

_ [«— zj [« -l. z] 

u — u will be always pofitive or always negative, according as the 
concavity or convexity is towards C. From whence it appears, 
that the value of [u — - IT . j (fuppofing both it and its reci¬ 
procal to be finite) will be pofitive or negative, according as the 
curve is concave or convex towards C. 




THE 


Chap. VI. 


ANALYSIS. 


6 7 





k.W«i^iWW*W^«i^WW^Wi 


THE 

RESIDUAL ANALYSIS. 


CHAP. VI. 

Of the Invefiigation of ufeful Theorems, by finding 
the nature of a Curve front a given property of its 
Tangents. 

I. 

Being parallel to AE j it is propofed to find a curve Fig. 20. 
line , Juch, that , ab being a tangent thereto , at any 

point thereof Bb] — Aa] (= Bb + Aa x Bb —Aa) 

Jhall always be equal to the invariable fquare c*: 
which tangent is fuppofed to interfeft AE and BF 
in the points a and b refpeblively. 

Suppofing p to be the point where ab touches the required 
curve, let mp be drawn parallel to AE ; and call AB, a ; Am, 
x mp, y . 

Then will Aa be =:y — x [* ^.y], 

Bb =y + ^—xx[xj-y], 
and Bb] — Aa] = a z — zax -f- 2 ay — c 1 . 

K 2 Hence, 




68 


THE RESIDUAL 

Hence, by refidual divifion, we get 

“ — 2** x [* J-^]‘ X [* ^y\ + ay [x ju y ] = o } 

from whence we have fx _i.vl = —-—. 

Which laft quantity being fubflituted for its equal in the value 
of c X y we find 

1 c* - 

y = - x 2 x — a. 

s a 

The curve therefore is a Parabola; AB coincides with a 
diameter thereof, the parameter of which is = —; and, V be- 

a 

ing the point where that diameter meets the curve, AV is == BV. 

Corollary. AB being a diameter of any conical Parabola, 
and aAa parallel to the correfponding ordinates; if Aa on one 
fide of A be equal to Aa taken on the contrary fide of A, the 

tangents ap, ap will always interfcd: each other in the right 
line Bb, being that ordinate continued whole abfcifla VB is 
= VA. 

2 . 

F»g. ai. BF being parallel to AE, it is propofed to find a curve line , fucb , 

that, ab being a tangent thereto, at any point thereof, Bb] -f- Aa] 
Jhall always be equal to the invariable fquare c x - 

Let the lines AB, A m, mp be as in the preceding article : 
Then Aa being =y — x [x _^y], and Bb =y + a — x x [# -i_y]; 
Bb] + Aa] is = 2 x y — x[x .Jy]| -f- a 1 — 2axx[x-^yY~\- 
2 ay[x -L.y] = c\ 

Hence, by refidual divifion, and dividing by 2 [*-^.y], we get 
2x z [x-Ly] — 2 xy + a 1 — 2 ax x [* ^ y] + <9 = 0 
from whence we have [x ^ y] =- ay —-. 

1 lax — ■« 2 ** 


Which 


Chap. VI. ANALYSIS. 

Which laft quantity being fubflituted for its equal in the value 
of c\ it appears that the equation of the curve is 

y' = J, X<2‘-: 

Anfweriqg to an Hyperbola j whereof CV equal to and pa¬ 
rallel to AE and BF, is a femi-diameter ; whofe femi-conjugate 
is AC = BC = jj C being the center. 

As by inveftigating the firft propolition, we difcovered a re¬ 
markable property of the Parabola; fo here we difcover a property 
°f the Hyperbola, equally remarkable : And it is obvious, that 
a Corollary may be drawn from what is done in this article, fimilar 
to that in the preceding. 


BF being fiill parallel to AE; it is proofed to find a curve line, Fi e . 
Jucb y that , ab being a tangent thereto , at any point thereof the 
rectangle Aa x Bb Jhall always be equal to the invariable fquare c\ 

The lines AB, Am, mp being as in the preceding articles ; 

Aa will be —y — x [x^y], and Bb = a [x^y] +}—x[x_Ly]. 

Therefore A a x Bb will be = a [x.-.y] xy — x[x_!_y] + 
y —x[x_^y]| =f ‘. 

Hence, by means of our refidual divition, we get 
a + 2x x y — x [x _jl/| — ax [* _£.y] = o ; 
from whence we have [a: _i_ / — ay + 2xy 


2 ox 2 ^ 

Confequently, by fubftitution, we find 

Bb = * and = , 

4 . ax + 


Aa = —2- 

2.(1 4 - 7.x * 


Therefore, in the firft cafe, the equation being/ = x 77 ^ 7 r 
the curve is a Circle or an Ellipfis; and in the fecond cafe, the 
equation being / = £. x ax + x\ the curve is an Hyperbola. 
AE, BF touch the conic fedlion, or oppofite hyperbolas, in A 

and 


THE RESIDUAL 

and B: AB being a diameter of the fe&ion; whofe conjugate 
is equal to zc, and parallel to thofe tangents. 

Corollary I. It appears that c , the fejni-conjugate to the 
diameter AB, is a mean proportional between Aa and Bb. 

Corollary II. x being to — as a to the value of Bb; 

it follows, that if AE, BF be parallel tangents to any ellipfis, or 
oppofite hyperbolas, and mp any ordinate to the diameter AB, 
a line drawn from the point of contadt A, fo as to bifedt mp, will 
always meet the tangent from p in the line BF. 

Corollary III. Let ab be another tangent to the curve. 

Then, Aa being to Bb as Aa to Bb, the right lines ba, ba will 
meet in the diameter AB, or in the continuation thereof. 

Scholium. If it be required to defcribe, to a given center, 
a conic fedtion, or oppofite hyperbolas, that fhall touch three 
right lines given by pofition : Draw a line parallel to one of the 
given tangents, fo that the given center fhall be between them 
and equidiftant from each, which line it is plain mull be another 
tangent to the required curve; and then the diameter correfpond- 
ing to the two points of contadt of the parallel tangents may be 
readily found by this Corollary, and its conjugate will be known 
by Cor. I. The defcription of the curve then eafily follows. 

Corollary IV. By confidering BF to be removed to an 
infinite diftance from AE, we may infer from the laft Corollary, 

that, if AE, ab, ab be tangents to any conical parabola, right 

lines drawn through a and a parallel to ab, ab refpedtively, will 
meet in the continuation of that diameter AB which paffes through 
the point where AE touches the curve. 

Scholium. This Corollary, it is obvious, enables us to de¬ 
fcribe, with great facility, a parabola that fhall touch three right 
lines given by pofition, and have its axis parallel to another right 
line given by pofition ; and likewife to defcribe a parabola, that 
fhall touch four right lines given by pofition. 


Chap. VI. 


ANALYSIS. 7I 

4 - 

AB, BC making any angle at B; it is prcpofed to find a curve Fig. 24. 
ZrW, fucK that , ab &v>zg 0 tangent thereto , <7/ any point thereof 25] 
rectangle Aa x bC Jhall always be equal to the invariable 
fquare c\ 


Suppofing p to be the point where ab touches the required 
curve, let AD, mp be drawn parallel to BC, AB refpetfivelv ; and 
call AB, a BC, b; Am , * ; mp , y. 

Then will Aa be =y — x[xly } 9 Ba = a —y— x [x±y ] 9 
Bb = 


- and bc =Aa 

0 -j] 

Therefore 


* —> —*l>~ y] 
[* — >] 


AaxbCis = J *«—.*[* . y-x[xj_ y ]\ 

l*—j] 

Hence, after multiplying by we. by means of our 

refidual divilion, readily find 


[x^ rl — “x + b—i-xyT^ 

L 2 bx - 1 


Confequently, by a proper fubftitution, it appears that the 
equation of the curve is 

ax 4- by -f-Tq = 4 ab 4I 4c 1 x xy : 
wh jch correfponds to an Ellipfis, or Hyperbola. And if CD be 
parallel to BAj AB, BC, CD, and DA will be tangents to the 
ellipfis, or oppofite hyperbolas. Moreover, taking AE and CF 

each equal to the right line EF will be a diameter of the 


Figure, whofe conjugate diameter will be = — x s/ab X c. 

b * 

Corollary. The middle point P of the right line AC is the 

center of the Figure: And, (fuppofing ab another tangent to the 

c “ I T e >) Aa being to Cb, as Aa to Cb; if ab, ab be joined, and 
thole lines bifected, a right line drawn through the points of 
bifedhon will pafs through the point P. 


For 


A 


72 THE RESIDUAL 

Fig. 26. For let Bd be to BC, as Aa to Cb; and parallel to Cd draw be, 
2 7 ‘ bf; alfo, having bifeded thofe three parallels in m, n, and 0, 

and the three lines AC, ab, ab in p, q , r refpedively, draw mp, 
nq , or, which will all three be parallel to AB. Then from the 
analogies 

de : Cb : : Bd : BC : : Aa : Cb, 


and df : Cb :: Bd : BC : : Aa : Cb, 
it appears that de will be = Aa, and df = Aa. 

Therefore, in cafe the firft (Fig. 26.) ae and af will each be 
equal to Ad ; and the parallels mp, nq, or each equal to 1-Ad. 
Confequently, mno being a right line, pqr is, in this cafe, a 
right line parallel thereto. 

Moreover, in cafe the fecond (Fig. 27.) ae and af will be 
equal to Ad -f- 2de and Ad 2df refpedively ; and the parallels 
mp, nq, or equal to ~Ad, yAd -(- de, and yAd -}• df refpedively. 
It is evident therefore, that, in this cafe, pqr is a right line 
parallel to dC. 

Scholium. If it be required to defcribe an ellipfis, or oppo- 
iite hyperbolas, that lhall touch five right lines given by pofition ; 
the center of the Figure may be eafily found by this Corollary: 
And then we may proceed according to the Scholium to Cor. 3. 
of the lafl: article. 


Fig. 28. 


AB, BC making any angle at B j it is propofed to find fuch a 
curve line , that , ab being a tangent thereto, at any point thereof, 
Aa x bC Jhall always be to Ba x Bb in the invariable ratio of a 1 
to c\ 


Lines being drawn and denominated as in the preceding 
article; we have, from what is there faid, and from the given 
ratio of the redanglc Aa x bC to the rcdangle Ba x Bb, 


c x xb[x^.y] — axy — x [xj.. y] +y — * = 

«*x o— J+ *l*— 3i\ • 


Hence, 


73 


Chap. VI. ANALYSIS. 

Hence, by means of our refidual divifion, (x — x being the 
divifor,) we get _ _ 

r «. be'y — a . 2a x — c x , x 4- 2a x — 2c* . xy 

IX 1 y I — —-—— - — -. 

L 2 bc x x + 2a 1 — 2c x .x x 


Which laft exprefiion being fubftituted for its equal in the 
equation above, the equation of the curve will be obtained. 
Or the fame may be found (perhaps more readily) by proceed¬ 
ing as follows. 

L et — l — be fubftituted for its equal [x ^y] t in the firft 

(y~*l . 

equation, and there will refult 


c' x b —' a [y x] x y [y x] — x + y [y~x] —xf = 

a x a — y . [y x] -f- xt . 

From whence, by refidual divifion, (y — y being the divifor,) 
we find 




bc x y — a . 2 a x — c* . x -4- 2 a 1 - — 2 c' . xy 
2a 4 — 2 a . 2 a x — c x . y -f- 2 a 1 — 2c*. jr* 


Therefore, [x-i. y] being = - 


bc % y— a . 2 a x — c x . x 2 a x — 2 c x . xy 


ly *] _ _ 

2a ♦ - 2a . 2 a x - c x . y 2g*— 2c*.>» 


- will be i_ - __—_ - 

2 bc x x 4- 2a x — 2 c* • x 1 £c*^ — a . 2a* - tf*. x 4- 2a*- 2f\ xy 

And the equation of the curve, from thence found, is 

a cx 1 + b'cy' -f- 2 ab . 2a — c 1 . xy — \a*bx = o; 
which correfponds to an Ellipfis or Hyperbola, according as c is 

f reater or lefs than a. —It appears that AB, and BC will touch the 
'igure in A and C : And that if CE be parallel to BA, and equal 

to c . a j AF, parallel to BE, will coincide with a diameter 
of the conic fe&ion ; which diameter will be equal to BE, 

1 C X V 3 4 

2a 1 


and its conjugate cq\ial to 


✓ c x c/3 a x 


Corollary I. This conclufion fuggefts fome remarkable 
properties of the conic fe&ions j and alfo an eafy method of de- 

L fcribing 


74 


the residual 

/bribing elliptical or hyperbolic trajectories, that /hall touch right 
g i y en by pofition, in certain cafes, to which the theorems^ 
the preceding articles are not fo readily, if at all applicable. 

Example. To defcribe a trajeBory that Jhatl touch three ritht 
lines given by pojitton, and two of them in given points. S 

F.g. 29. Let AB, BC, ab be the three given lines, and A and C the 
points of contaft of the two firft of thofe lines. 

, U V, e ‘ir® ? A . take Ae e <jual to AB, and parallel thereto 
", S'* alfo > h . avin g drawn eb/j draw /Bg meeting Cag in g: 
then will gA continued coincide with a diameter of the required 
trajectory In the fame manner may the direction of the diameter 
try 1 C -.l f ° und . ; and confequently the center of the conic 

TUu 8 r the r P ? mt where thofe diameters interfeCt each 
other. The bufinefs then may be eafily completed, after the 
manner commonly taught by the writers on Conics. 

Th.sconftruCtion is fo eafily inferred from what is done above, 
that 1 think it unneceflary to be more explicit. 

r ' e ' 28 h C°r L K A f V [ f c be e 1 uaI t0 a ' required curve will 
be a Parabola. It therefore evidently follows, that, any two 
angents AB, BC being drawn to any conical parabola ApC, 
ouc mg the fame at A and C, and interlefting each other at B- 
th i! rd tangCnt ab be draw n (to the fame parabola) inter- 
2 hf-R tang ^ tS " 4 and b refpeCtively, AaxCb 

oertv of il a X 1 And L 3 know ledge of this remarkable pro- 
P y ie parabola enables us to find Tome others, and readily 

Art ? 3 ° rm thC bufiTlefs mention ed in the Scholium to Cor. 4. 

Elutions to thefe problems being eafily obtained by means 
our refidual divifion, and the firft theorem in the preceding 

re P ^^°“; 7 farthfr knowled ge of our doCtrine; they 
are, it is prefumed, not improperly inferred here.—In a fubfe- 

Cafculus ha f ’ W ,l Aa c fbeW how ’ , b y a different artifice in our 
inuedicnfed 0 ? 16 ° • Cr tbeorems relating to curve lines may be 
mveftigated from given properties of their tangents. 

ft ,T C °* tb fJ beorems here inveftigated may be feen, demon- 
£ Vnc ZZ™™' " &> ISAAC NEW TON’r Philqf. 


THE 


Chap. VII. ANALYSIS. 


75 


k 
* 


CT 


IDM&zMai 


m 




THE 

RESIDUAL ANALYSIS. 


CHAP. VII. 

Of the Evolution and Curvature of Lines ; with fane 
Inferences relating to the Focufes of refletted and 
ref ratted rays y and the curves call'd Cauftics. 


Perfe&ly flexible thread, dCdA, being applied along Fig. 30. 

A »Lg t h e convexity of the curve dCa , from d to a, fup- 
jtrg pofe the part a A (of fuch thread) to be extended in 
^C)0()eC)eCSC a right line that touches the faid curve in a ; fuppofe 
alfb, that, whilft one end of the thread remains fixed 
at d y the other end A be moved towards D, (in the fame plane 
with the curve,) fo that the thread be continually unwrapped 
from the curve, and the part CP which is difengaged therefrom 
be always extended in a right line that touches the curve : then 
fhall the point A trace the involute curve APD that is faid to be 
defcribed by the evolution of aCd, which is itfelf called the 
evolute , and the right line CP is called the radius of evolution cor- 
refponding to the point P. 


2 . 

Let P r be drawn at right angles to CP; and, with the radius 
CP, defcribe the circle EPF. Draw any other radius of evolu- 

L 2 tion 


THE RESIDUAL 

tion eq, e being between a and C; join Ce ; and draw Cqr, in¬ 
terfering Pr in r. Then, by the nature of the evolution, eqX-the 
curve Ce being manifcftly — CP, eq + the chord Ce is id's than 
CP; confequentiy the right line C q is dill lefs than CP. But Cr is 
greater than CP : therefore Cq is lels than Cr, and the involute 
AP is wholly between the right line Pr and the evolute aC. 

The point e being on the other fide of C, and g being the 
point where PC continued interfefts er 5 eq (= CP -\-lhe curve 
Qe ) * s ^ an e S “bifP : therefore gq is lefs than gP. But gr is 
greater than £p : therefore gq is lefs than gr , and the involute 
Py is wholly between the right line Pr and the evolute Ce. 
Consequently both Pr,. and the circle EPF, touch the involute 
in P. 

The point e being between C and d , fuppofe F to be the point 
where the circle EPF interfeds eqr. Then cq (= C?-\-the curve 
Ce = CF-f the curve Qe) will he greater than eF: confequentiy the 
circular arc PF falls within the involute P^; and it is plain, that no 
circle defcribed through P, with a radius lefs than CP, can pals 
between PF and P q. Moreover, gq being lefs than gP, (as before 
©bferved,) a circle defcribed from any point g through P, with 
a radius ^P greater than CP, paffes without P q and PF. There¬ 
fore no circle defcribed through P can pafs between P q and PF. 

The point e being between C and a , CP (as is proved above) 
is greater than C q : confequentiy the circular arc PE falls without 
the involute P q * and it is evident, that no circle defcribed through 
F, with a radius greater than CP, can pafs between PE and P q. 
Moreover, J being the point where qe continued interfetf s CP, 
xLr/° "b* f ^ e curve eC) being lefs than qf-\- /'C, P f is 

lefs than qf: confequentiy a circle defcribed from any point f 
through P, with a radius f P lefs than CP, pafles within P q and 
FE. It follows therefore, that no circle defcribed through P 
can pafs between the circle EPF and the involute curve APD, 
an other circles palling either within or without both the faid 
circle and curve. 


3 - 

The circle which touches a curve fo clofely, that no circle can 
be drawn through the point of contact between them, is faid 
to have the fame curvature with the curve at that point Which 

circle 


Chap. VII. ANALYSIS. 

circle is called the circle of curvature ; its center, the center of 
curvature ; and its femidiameter, the radius of curvature, corre- 
lponding to fuch point of contadh 
It appears then, that the circle EPF (whofe center is C) is 
the circle of curvature of the involute APD, at P. Therefore, 
by confidering any curve as an involute, the radius of curvature 
(or evolution), at any point thereof, may be readily found, as in 
the following articles. 


The curve APD being referred to a bafe AK, fuppofe aCd Fig. 
to be the curve by whofe evolution APD is defcribed : draw CH 
paraUel to KA ; and fuppofe GH, parallel to the ordinate PB, to 
mterfedt the faid bafe in I Call AB, BP, y ; the curve AP, 
the tangent NP, /; the fubtangent BN, j; AI, 6 ; GI, c ; 
OH, CH, zi; the curve aC , w ; and (CP) the radius of 
curvature at P, R. 

Then, by fimilar triangles and Chap. 5. Art. 5. 

, . [y^ rj (_ [y ^ 

Therefore i is = 1 x [v R], and y — c = v. 

From the laft equation, we get, by refidual divifion, 

4- . X J- R? 

Hence R is found = -- **£ v ~ ^ r f* — rJ 

by Chapter 2. Article 8. 

[x _Z_ zlJ 

— - h±y} x [±±>] ™ 

[x-uz] [ x z y • 1 bereiore, by fubfhtution, 

we have 

R — [l±jJ * [«-*-*]* _ 

[*■*■*] 

where « _ Si+[x^}y. 


From 


7 8 


* 


THE RESIDUAL 

From thofc equations, and what is proved in Chap. 2. Art. 9. 
other exprefiions for the value of R may be obtained, viz. 

r — f? — x O— __ rr±^i!. 

[y — z] [y ~<T 

and R = = _ IS_±d. 

[z -a- x] O 

Or the fame may be obtained from thefe analogies, viz. 
y : t :: u + b — x : R :: [v 2-ii\ : [v R] (= [v -1- w ]) 9 
which follow from fimilar triangles and Chap. 5. Art. 5. 

Example. In the Parabola , being =y, * is = ~, 
O ^ *] = % [y -a. x] = and [yjLZ ] (= v^r+^TTlJ 1 ) 
= J1 4- Therefore R = ^ is, in fuch curve, 

equal to 


In the Ellipfis and Hyperbola 2amx 4- ax' is = 2 my *: and by 
refidual divilion, and fubftitution, 


R is found = - 


* 4 - 4 «y 4 - 

2 a % ni ' 


a being the Parameter, and m the femi-tranfverfe Axis. 


It is obfervable, that, in each of the Conic Sections, the 
Radius of Curvature, at the Vertex of the Figure, is equal to 
Half the Parameter, 


5 - 

COROLLARY. 

The equation of the involute curve being given, we may, 
from thence and what is done above, readily find the nature of 
the evolute. For, from what is faid in the lafl article, we have 



1 "X — z~] 

= — 7 — ~r — y — c > 

[x 22 y] 

i> *-*] ' 


u 


Chap. VII. 


79 


ANALYSIS. 

_ L? z 3 I y ^^ L 

[?■*-*] 

—_ t*±}]x[*±*L i _ i 
[*—/] T 

From whence, by means of the given equation, x and y may 
be expunged : Confequently the relation between v and u will 
then appear. 

Example. Let the involute he the coniccd Parabola 1 wbofe 
equation is ax = y\ 

Then [y-i-x] being = 2 , [y-a.*] = and [yu.z]'z= 1 + £j. 
v is = — Cy and u = — -f- *£ _ 

Hence y = = EEE±^f 

4 I 3 I 

Suppofe c = o, and b = {a ; then ^ = fF)', 
and confequently = u\ 

Therefore the evolute aCd is, in this example, the femicubi- 

cal Parabola : And AI being = ~ r and G coinciding with the 

point I, that point is the center of curvature correlponding to 
the vertex A of the Parabola APB. 


The curve APD being of the fpiral kind, whofe ordinates all Fig. 32. 
ifliie from the point G, fuppofe aCd to be the curve by whole 
evolution APD is defcribed r join CG, CP ; and draw GH, 

n e ^ ndlCU i? r t0 CP and the tan g er »t PO refpe&ively. 

Call GP y j GQ, (= HP,) p ; the curve aCdy w 5 and (CP) 
the radius of curvature at P, R. 

' Then, CH being = R — p, and GH = </y — p\ GC will 

= \/R a — 2y>R y\ Now, by Corollary to Article 8. 

Chap. 5. 

CH 


8o 


THE RESIDUAL 

CH : CG y/R 1 — 2 j>R+y] : [uj.w] (— [b-j.R|) 

* i. e. R — p : */R‘ — 2/>R -f y L 

*• R[«—R3—— +rO—>] . r-# , pi 

VR* — 2/>R + y' 1 . 

From whence it is evident, that 

y[ u — y\ — R [« ^ ^] is =r O ; 

and confequently R = - y u being any fundion df y. 

[« — />] 

Example. Suppofe APD to be the Logarithmic Spiral ; and 
s the fine of the invariable angle GPQ, made by the ray GP 
and tangent PQ> radius being unity. 

Then GQ^(= p) will be = sy, and confequently R = * ^ u ~ y ^ 
_ y_ l u — Pl 

s 

Corollary. CP (= R) being = and / = ty; GH will 

be = \/y L — sf =^/i — s l x y, and CG = s/ ~— y % = 
>/1 —- P x L . Therefore the fine of the angle GCH will 

always be = s; and confequently, in this example, the evolute 
aCd is the fame curve the involute APD, but placed in a 
different pofition. 

7 - 

Fig. 33. Suppofe cpq to be a kind of cycloid defcribed by the point p 
carried about with the curve aB revolving along the immoveable 
circular arc AB, as in Art. io. Chap. 5. 

From D, the center of the immoveable circular arc, draw D^ 
perpendicular to />B continued ; and, to the continuation of DB, 
draw the perpendicular pc. Call the radius DB, R; B/>, #5 
Vp, y s Bg, w ; and let fp , the radius of curvature of the 
cycloid at p> be called E. Then the tangent pt being (by 
Chap. 5. Art. 10.) parallel to ^D, D/(=/>) the perpendicular 
from D on that tangent will be = w -f- x. Moreover, R being 

to 



Chap. VII. 


ANALYSIS. Si 

to w as * to Be (= 22), ^l* will be = x ‘— ^1, and 

/ = = R+II+ *• - = R* + * + a«*. 

Therefore E (= - 1 — by the laft Article) is equal to 

x \ji — jcJ -|— IV [u-^-x] + X [u -Z- w] 

[u zv] 4 [« — x] 


Scholium. If R be infinite, i. e. if AB (inftead of being a Fig. 34. 
curve) be a right line: Then Ag being perpendicular on pB ; 
and B^, Bp, Ap, and AB being called w , x , y , and z refpec- 
tively ; we have p (the perpendicular from A on the tangent to 
the cycloid at the point p) = x — <w, and y z = A^| -j- pg\ = 

Z — w z -f- x — «z£;| — 2* + x 1 — 2 wx. Therefore, in this 

cafe, E is = 

2 . [u _i_ jc —- «/] 

Example I. aB be a circular arc whofe center is d ; and Fig. 33. 
call the radius Bd , r; dp, Then, — Be] being = ~cp\ = 

£ — Be — r| = £ — r*-|-2rx Be — Be] , it is evident that Be, 
or its equal is — — r f . Hence w = — X x z -4- r" — 

Confequently p (= w -j- x ) is = 2 -. * -f . x~~\ 

/ = R* + 1.^7 + E±I.^ and 

g _ 2 . R 4 r • * 3 _ R 4 r • 

R 4 2r .a: 1 — R . r 1 — R 4 r . a — rw‘ 


Corollary I. Suppofe the defcribing point p to be in the 
periphery of the moveable circle. Then, ^ being — r, p is 

= 5Jir.* andE 

2 r 3 s * r R 4 2r 


M 


From 


82 


* 


THE RESIDUAL 


From whence we have y z = R* -f- p l ■ 




./>% and 


v/E" — 2/>E+> ,l = Df = y (the ray from D to the point of 
the evolute correfponding to the point p of the involute) == 

J R l — R ‘ R r ~ x . Now Dp - = p being = \/R~ — w* 
_ r . R + ?,r| 6 r 

= J R s 


X* is = 4r* — and confequcntly 

R + , "z R a 7 ;z 


4 r 

+i> a 


R + arf ' ' R -f* 2r i ^ 

Bringing the laft expreflion to the fame form with the value 

of y 1 we have 'y = R* -\-p l — ■ R -^ p . p\ where R 1 mud 


be: 


R 4 


R + ir 


- and 


R r 


& + 2^1 


R* 

R + 2rV* 


Therefore, R is = 


-- , _^-. and R - 4 - 2r = R. Hence it is plain, 

R + 2 r* R + 2 r* ' . 

that the evolute of the cycloid opq is another cycloid, which 
may be defcribed by a point in the periphery of a circle 


whofe radius is 


Rr 

R + 2 r 
R 


revolving upon the immoveable circle 


whofe radius is g—and center D ; and the pofition of the 
evolute with refpedt to the involute is very obvious. 


Corollary II. Let the fixed point P be fo fituated, that 
AP and DP fhall be refpe<ftively equal to a p> dp j and let r be 
fuppofed equal to R. Then, the arcs AB, aB being equal, if the 
ray PB be drawn, the angle DBP will always be equal to the 
angle DBg\ Therefore the evolute of the cycloid opq is the 
cauftic by reflection of the circle AB, P being the focus of the 
incident rays. 

Now, R being = r, E will be = ~ and 

¥>f — E — x (the difiance of the focus of the reflected rays from 

the 


Chap. VII. 


ANALYSIS. S3 

the point of incidence) = where 

x is = PB, and £ — PD. 

Hence it is obvious, that if £ be = r, i. e. if the focus of the 
incident rays be any where in the periphery of the circle AB, 

the focal diftance B f will always be = ^ 

Corollary III. Let r and ^ be fuppofed each equal to -J-R. Fig. 35. 
Then, the arcs AB, aB being equal, and the defcribing point 
p coinciding with a, if any ray PB be drawn perpendicular to 
AD, the angles DBP, DB/" will always be equal. 

Therefore the evolute of the cycloid opq is the cauAic by 
reflection of the circle AB, the incident rays being parallel. 

Now, ^-R being = r = ^, E will be = {x and the focal 
diAance Bf == = {-w. 

Corollary IV. AB being any reflecting curve whatever 5 
the focus of rays reflected from any point (B) thereof may be 
found by the above theorems, by confidering R as the radius, 
and D as the center of curvature of fuch curve at the point of 
incidence. 

Example II. Let ap be perpendicular to the tangent to the Fig. 36. 
moveable curve at a, and the points P, A, D in a right line : 
and fuppofe the curve aB of fuch a nature , tkat y the arc aB being 
equal to the arc AB, pB (= x) Jhall always be equal to n x PB, 
n being invariable . 

Then PB being continued, and Dh drawn perpendicular 

thereto, wv/R" — will be = \/R* — w\ v being put for 
Bh: for it follows from Chap. 5. Art. 8. that the Anes of the 
angles DBh, DBg will be to each other in the invariable ratio 
of 1 to n respectively j thofe fines being refpeCtively equal to the 
cofine of the angle made by PB and the common tangent to the 
two curves at B, and the cofine of the angle made by pB and 

the fame tangent.—Moreover, calling AP, d ; d + Rj — R 1 

4“ v 1 will be = v means of which equations 

M2 [k 


*4 


THE RESIDUAL 

[u ±. w] and [u x] may be eafily expunged out of the equation 
k — - - - z j i---5 and then we fhall have 

[u s. w] 4- [u -i. X ] 

p to — w-}- w 7 , — n l v*. x 

iv — nv.x — rfv x 

Corollary I. The fines of the angles DBh, DBg being 
to each other in the invariable ratio of i to n refpe&ively, the 
evolute of the curve cpq is the cauflic by refraction of the 
circle AB, P being the focus of the incident rays, and i to n the 
ratio of the fine ot incidence to that of refraCtion. Confequently, 
taking x from the value of E, and writing ny inftead of x; 
B/, the diftance of the focus of the refraCted rays from the 

point of incidence, will be found =r-^_ , where v 

wy — nvy — nv x * 

is equal to PB. 


Corollary II. AB being any refracting curve whatever, 
the focus of rays refracted at any point (B) thereof may be 
found by the theorem in the preceding Corollary, by confidering 
R as the radius, and D as the center of curvature of fuch curve 
at the point of incidence. 

I" Scholium. If R be infinite, i. e. if AB (inftead of being a 
curve) be a right line, to which AP is perpendicular. Then, 
Ah, Ag being refpeftively perpendicular to BP, B/ j and AP,’ 
Ph, Bg, Bp, BP, and AB being called d, v, iv, x t y, and z 
refpeaively ; we have / — d' = = vy, x = ny, and 

w z=z nv. From whence we get x — w = ry _ nv = ny _ 

ny'—nd 1 nd 1 , » , * x „ . . x 

J, -— > and z -f- x —— 2 wx = y — d -J --ny — 2 n vy 

=/ — ny x — (T -J- n z d\ 

Therefore, by the Scholium to Article 7. E is — 

Confequently, in this cafe, by what is obferved in Corollary j. 
E — x, the diflance of the focus of the refra&ed rays from the 

point of incidence B, is = — ' £ — ny, P being the focus 
of the incident rays. 


8 . To 


Chap. VII. 


85 


ANALYSIS. 

8 . 

To find the point F where any refraded ray BF meets the Fig. 37. 
axis AF of any refrading curve AB. Draw the incident ray PB, 
the ordinate BC, the tangent B/, and BD perpendicular to that 
tangent. Call the abfciffa AC, x ; the ordinate BC, y ; the 
curve AB, z ; PB, v ; BF, w ; AP, d\ AF, e: and let the fine 
of incidence be to that of refradion, as 1 to n. Then, the fub- 
tangent, ordinate, and tangent being to each other as 1, [ x^y ], 
and [* _l z ] refpedively, by Chap. 5. Art. 5. we have 
1 : [x~y] : : y : y[x-^-y] — CD ; and (radius being 1) 


[*_lz] : i : : i : 


[* — Z ] 


, the fine of the angle /BC = BDC 5 


[at z] : 1 : : [x _£_ y] : ———, the cofine of the fame angle. 

[xJ-x] 

Moreover v : 1 : : d-\- x : the fine of the angle PBC; 

and v : 1 : : y : — the cofine of PBC. 

J V 

Now the cofine of the difference of two angles being equal 
to the redangle of the two fines plus the redangle of the two 

cofines of thofe two angles, —(- y ^~ ^ is the cofine 

v[x-z-z] v[x^z]' 

of the angle /BP, or the fine of the angle of incidence. 

Therefore, n times this Iafi: fine being the fine of the angle 
of refradion DBF, we have 
d x 4- y[x — y] 

— [x-i-z] 


e —x— y[x— y] (=DF) : w. 


Confequently, v being y z -\-d-\-x\ , and w ~ y/rsr, 

* X L +*+>[*-•->3 is _ < — * — y [* — >3 . 

T+x\ s/~e — ,v| 

by means of which equation, and the equation of the given 
refrading curve, e may be readily determined. 

Scholium I. If d be infinite, i. e. if the incident ray PB be 

parallel to the axis of the curve, the fine of PBC will be = 1, 

and 


86 


THE RESIDUAL 

and the cofine = o. Therefore, in that cafe, n : i :: DF : BF, 


and confequentiy n 


e T*—y I 


Scholium II. That the refrafted rays may be all parallel 
the angle made by BD and any ray after refradion at B muli 

be equal to the angle BDC. Therefore n x i+l+lll ±M 

T/ -i_ z] 

mud: be ==■ —-—-■ ; and confequentiy, in this cafe, 

»x~/ll^ ,] mult be = 1. 

Example I. The re fra fling c urve being Descartes’ Oval, 
uhofe equation is ‘_j _„ a = i ‘-f-/ ; 

we, by refidual divilion, get * + *+>[* __ b—x—y[x-i-y] 

*|~ -f- y* y/b — x \* 4- y x 

Which equation being compared with the equation 
n ^ X +y[* — yl e — x—y , 

s/7T^T? above found > * a PP ears 

that the fines of incidence and refradion being as i to n if 
[n being lefs than i, and a and b pofitive quantities,) dbe 
_ a, f will always be; == b ; and the refraded rays, from every 
point of the curve will all meet the axis in one and the fame 

aberration theref ° re wil1 be dleir comm ™ focus, without any 

Example II. Suppofe the curve AB to be a circle, uhofe 

T a U u *r^*n. Tben ’ / being = 2 ax — at 1 , yhc^y] is =a — x: 
and by fubftituting thefe values of y 1 andy [x^y] for their refpec- 

tive equals in the equation n x x frj[p_f-y] = e — x— y [x-£- y ] 

_ y/T+'X + f — ’ 

we have - ” : a + d e — a r , 

/—-===== = - 7- • irom whence, 

Vd-\-2.a-\-d.x vV -{- 2 . a — e . x 
when a , d , n , and x are given, e may be readily found; and, 
consequently, the aberration of any ray from the focus of rays 
falling on or very near the vertex A. 


From 


87 


Chap. VII. ANALYSIS. 


. ✓ </* + 2 . fl +d. X 

From the lalt equation we get • r= - = 

V* 1 - 


« . a -f- d 


2 . — e . * 


therefore ^ £1+^1+^ ; s _ *!L1±£, b being of any 

v/* 4 + 2 .a — e.x 

value whatever. Now fuppofing e = b x d\ and a — e = 
xa 4- d , ' a - it is evident, muft be = i, let a: be what 

it will: From which equations we have e ■= bd = — nd = 
n -U i , a and d = — —t-ij. Hence it appears, that if P be 
on the concave fide of AB, and its diftance from A be = — a > 

the refradted rays from every point of the curve will all diverge 
from one point in the axis, the diftance of which point from 

A will be = n -f- i. a. 

And it follows, that rays converging to P, (fituated as juft 
now mentioned,) and falling on the convexity of the circle,will all 

be refradted to F, AF being = n + 1 . a. 

Example III. Let AB be an ellipfis, whofe equation is y = 
ax —. a being the parameter , and b the tranfverfe axis . 

Then will — g — x] -\-y 1 be ~ — g — *1 + *** J x > 
g being of any value whatever. Hence, by refidual divilion. 


we get 


x — y[x -L. 


g — x — ±a + —x 




71 +/ 


Js- ! 


By compar- 


+ " —- 


ing this equation with that in Schol. i. it appears that, the in¬ 
cident rays being parallel to the axis of the curve, e will always 
be equal to the invariable quantity g, let x be what it will, if 






I—. *1 +< 


be always = n, or g — {a — — i 


always = n x g ■ 


-J- eix ■ 


5 i. e, 


• S - i"! + 


2 . 



88 


THE RESIDUAL 

2 • g — \ a • -i . x y — i| . x 1 always = ng x -J- 

** * a 2 g . x 7 i, y —— i . *\ Now, that this laft may be a 

true equation, x bei ng of a ny value whatever, g -*4* muft be 

= n% &'* 2 -g — i a • j —i = n .a — 2 g (=—2**•£—">), 

an d J — 1 1 = — n ‘ ~i — i* And from thefe equations we 
^ ave T = 1 — an ^ g = x —-— = \b x i + n- y which 

0 I — n 

value of g is the diftance of the remoter focus of the ellipfis from 
the vertex, when ~ is = l — n . It is evident therefore, that, ~ 

being = i — n\ and the incident rays being parallel to the 
tranfverfe axis, and falling on the convexity of the curve, the 
refracted rays from every point of the figure will, without any 
aberration, all converge to the focus juft now mentioned. 

And (reverfing the ratio of the fines of incidence and refraftion) 

it is obvious that, ~ being = - y-? , and the incident rays ilfuing 

from one of the focules of the ellipfis and falling on the con¬ 
cavity of the farther half thereof, the rays after refra<5tion will all 
proceed in a diredtion parallel to the tranfverfe axis. 

Writing — b inftead of b , the equation of the curve becomes 

y ~ 4X96 4 ~ j x » correlponding to an hyperbola whofe parameter 
is a , and tranfverfe axis b. It follows therefore, that, y being 

~ 11 1 j a nd the incident rays being parallel to the tranverfe 

axis, and falling on the concavity of the curve AB, now fuppofed 
an hyperbola, the refradted rays will all converge to the focus of 
the conjugate hyperbola. 

And (reverfing the ratio of the fines of incidence and refradtion) 
it is obvious that, ~ being = I__l, and the incident rays iflu- 

ing from the focus of the conjugate hyperbola, and falling on the 
convexity of the hyperbola AB, the rays, after refradtion at the 
laft-mentioned curve, will, from every point thereof, proceed in 
a diredtion parallel to the tranfverfe axis. 

THE 


Chap. VIII. ANALYSIS. 


89 


§ GOWOtKMJC I 

THE 

RESIDUAL ANALYSIS. 

CHAP. VIII. 

Of the greateft and leaft Ordinates, the Points 
of contrary flexion and reflexion, and the double 
and triple See. Points of curv r e Lines. 


I. 

y being fome fun&ion of x , y be always lefs or 

1 always greater than y , when a-, taken between cer- 

&)5()eOe()6( ta j n j s e i t h er j e f s or g rea t er than x, how near 

foever x be taken to x; the value of y , or the quantity denoted 

thereby, is confidered as a maximum or minimum , without regard 
to any value it may have when x is not taken between the faid 
limits. 


2 . 

The ordinate from a point of a curve is a maximum , or mini - 
mum , when, the curve being immediately continued on both 
fides thereof, it is greater, or lefs, than the ordinates which may 
be drawn, on either fide of it, from the adjoining parts of the 
curve. 


N 


3- Sup- 


90 


THE RESIDUAL 
3 - 

h Suppofe y to be the ordinate of a curve correfponding to 

the abfcifia x \ and y another ordinate (of the fame curve) 

correfponding to the abfcifia x. • Then if y be a maximum , it 
will be greater than y , whether x be lefs or greater than x y pro¬ 
vided x be taken between certain limits: and confequently the 
refidual y — y will, in iuch cafe, be always pofitive, when .v 
is taken either lefs or greater than x t between thofe limits. 

But if y be a minimum , y — y will be always negative, when 
x is taken either lefs or greater than *, between certain limits. 

Now feeing that, when y is a maximum or minimum , the 
value of the refidual y — ^ mud be always pofitive or always 

negative, when x is taken either lefs or greater than x , between 
certain limits : it is obvious, that, in fuch cafe, the value of 
y — y -r~ x — x will, accordingly, be pofitive when x is lefs 

than x , and negative when x is greater than xi or pofitive 
when x is greater than x , and negative when x i6 lefs than x, 
how near foever x be taken to x . Therefore, by Chap. 4. 
Art. 1. 

[x _i_ y] will then be = o, or —-— = 0. 

f *~y] 

Moreover, fuppofing b to denote a value of x in either of 
thofe equations, and d the correipondent value of y, d being a 

maximum or a minimum , will, accordingly, be pofitive when 

X u 

x is lefs than b , and negative when x is greater than b ; or pofi- 
live when x is greater than b , and negative when x is lefs than b t 


9 * 


Chap. VIII. ANALYSIS. 


how near foever * be taken to b, between certain limits. There¬ 
fore, by what is faid in Chap. 4. Art. 3. that expreflion, viz. 


-—j will be = x — T>\ x Q 

X - 0 


m being fuppofed an odd number, or a fradlion whofe numerator 
and denominator are both odd numbers > and Qjfome algebraic 
expreflion confiding of fuch quantities, that its value fhall be 
real when x is either lefs or greater than b , (between certain 
limits,) and that neither it nor its reciprocal (hall vanifli when at 
is equal to b. . 

And it likewife follows from the article laft mentioned, that, 
the abfeifla being equal to b t the correfpondent ordinate fhall 
be a maximum or a minimum , according as the value ot 
when x is equal to b , is negative or pofitive, m being as juft now 
fpecified, But, if m comes out contrary to our fuppofition, the 
abfeifla, when equal to b , will not correfpond to an ordinate that 
is either a maximum or a minimum.— This we fhall, by and by, 
explain by proper examples. 

The equation == * — b\ m x Q^being divided by x — b\” y 
we have • y = Q ; where (d being finite) m -f -1 muff 


be po-fltivc, otherwife j and cannot be finite. Now, by 


Chap. 2. Cor. to Art. 6. the value of . = nm+ l* w b en x is = b, 


is equal to the value of the quotient of [x divided by 
m-\-i . x — b\ n > when x is taken — b: Therefore f will be 
equal to the value of =& 44 =-, when x is taken as Juft 

now mentioned. Hence it is evident, that m and q may be found 
by refolving [x _s_ y] into two fuch factors, (F and G,) that 
one of them (F) fhall be fome power of x — b , and the other 
(G) fhall neither vanifh nor become infinite when x is therein* 

taken equal to b: for by comparing x — b\ m with F, m will be 
known ; and a will be the value of —when x is equal to k 

Moreover, Q will be real or imaginary when x is taken Iefs- 

N z os 


92 


THE RESIDUAL 

or greater than b y between certain limits j according as G is reel 
or imaginary when x is fo taken. Therefore, by means of the 
faid factors F and G, we may readily know whether the ordi¬ 
nate correfponding to the abfcifTa b be a maximum or minimum , 
without afligning the general value of Q. 

Example I. Suppofe y = ax — x 2 , a being invariable. 

Then we (hall have [xj~y] —a — zx =z x — - x—-2 — o- 

2 * 

where x is = 

2 

Now, b being — we have F = a; — j = x — S\ m y and 

G = —2. Therefore, m being = 1, and q (the value of — 

when * is — b) being a negative quantity, y is a maximum 

when ac is = —. 

2 

Example II. Suppofe y = x 4 — a’x, where a is invariable . 
Then we have 

[* -y] — 4 *' — a' — * — ^ X 4,V + + 4*’ = o ; 

where the real value of x is . 

4* 

Now, b being = 4 , we have F = * _ - — x— bf, and 

4 J 4» 1 1 

G = 4 l a -f* 4 'ax ^x\ Therefore, m being = i, and q 
(the value of -- ^ when x is = b) being a pofitive quantity, 

y is a minimum when x is = . 

' 4* 

Example III. Suppofe y = x l + a T — x ; |s a « 

Invariable pofitive quantity . 

Then we fhall have 

-.1 _ 2*xa } —*^—2** , 8a 5 *— 16 ** 

1 ( - 

* 2* Xa*— — 2* 1 being fuppofed z=z g, 2* x a 3 — a 3 [I will be zz g 2 *% 

and 


93 


Chap. VIII. ANALYSIS. 


—- 8a- X * -» —l X 2$ a x -f 2 *ax + 2x* 


■ ^ n i * -j- /.-a* “r *•» . r i 

21 , £ being put for the nu- 

1 — X * + 6g + I2* x 


2AT X « J — * 3 | 


^ I I 

merator zx x —• x } | r — 2*‘) = o ; and - = 

[* j -y] 

_ a 3 — X 3 1 * _ _ * <?j* x q r 4 - ax 4 - ** 1 * o 

2AT X fl‘ * 3 |^ — 2** m 2AC* — 2x X a 3 — * 3 j^ 

In the equation [x y] = o , the real values of a; are o and 

4 ; and, in the equation ■ = 0, the real value of x is . 

2 1 j] 

Taking £ equal to o , we have F = a; — 0 = * — and 

— 8 X a- — X 2*<a l 4- 2^ajc -f- 2 x z 

G — — — — -■ —. Therefore, tn being ma- 

a 3 '— a* 3 ] X g*x— 4 -f- bg -f-1 2 x* 

G • • 

nifeftly = 1, and q (the value of ——- when x is = b) being 
a pofitive quantity, y is a minimum when x is = 0 . 

Taking b equal to we have F == x — ^ = a* — £| w , 

and G = . ~~ 8 * X23 a + 2^*-J- _2*__ Therefore, w being = 1 

and q being a negative quantity, y is a maximum when x is = 

Taking b equal to a, we have F = x — a\ T = x .— b\ m > 

and G = Therefore m being = —■ —, and 

a - + ax + x \i b 3 

q being a pofitive quantity, y is a minimum when x is = a. 

and 8a 3 * 3 — 1 6x 6 = g 3 -f bg*x* 4. i2gx+. From whence it is evident, that 
if g be multiplied by g z 4- 6 gx z 4* I2x 4 , the produft will be equal to 

8<?V — i6y 6 j and therefore it is plain that - * * a * ^ is equal 

a 3 —**|T . 

to __ 8a 3 * 3 — 16** __ __ _ S^x rfi** _ 

ai r- -f bgx 1 4- 12** a % — 'jc 3 |^x g'x—i 4- bg 4- 12^ 

numerator and denominator being each divided by *% the variable quantity by 
which both 8aV* •— 16* 6 and g 1 -f- bgx* -{- i2* 4 are divifible. 

Exampis 


9 + 


k 


THE RESIDUAL 

Example IV. Suppofe y = a 4 x — aV — *aV Jax 4 
jx\ where a is an invariable pofitive quantity . 

Then will be = £ 4 — 2 a'x — 2 a*x* -f- 6**’ — = 

— 3 x x-- x*x -|—x x — a\ =zo-, 

S3 V 3 

where x is equal to - ■ } or * ** ■ j or <?. 

v 3 V 3 


Taking ^ equal to we have F z=x -^ = x — b\ m i 

_ S3 _ ^ S3 

and G '■ = — 3 x x -(- —= x x — a\ . Therefore, m being 
evidently .= i, and q (the value of -when x is = b) being 
a negative quantity, y is a maximum when a: is = -^=r. 

v -i 


Taking b equal to-we 

v 3 


have F = x -f- = x — b\ m , 

S3 


and G = — 3 x x - ^ x x — a\ . Therefore, m being =1, 

S3 

and q being a pofitive quantity, y is a minimum when x is =--~=. 

S 3 

Taking b equal to a, we have F — x — a] = x — b\ m , and 

G = — 3 x jf- - x x 4 -Therefore m is = 2 j 

S3 S3 

which being contrary to our fuppofition, it follows, that y is 
neither a maximum nor a minimum when x is = a, notwith¬ 
standing this value of x is determined in the fame manner as are 
the other two values above fpecified. 


Example V. Suppofe y = x -f* a — x [ J , where a is an 
invariable pofitive quantity. 


Then we have [x a.y] 



a h — 2a r x* 


* Xl* + 3S**+3** 


CffAP.-VIIf, ANALYSIS. 


91 


— X — 2 i « 5 — 2 1 a*x — 20 s * 1, 

23 _ 

a* — at 3 ) 1 X g 1 + 3£** + 3* 4 


»g being put for a i — x l ) = Oi 


and 


l*^y] 


TE3l _ x — a\*XS : + ax 4 - * a l*_ ^ 

-?n}_ > --- ■' — 


a 3 r- xV — x* 


In the equation [x ^.y] = o, the real value of a* is and 

27 

in the equation •—— =•p, the real value of x is a. 

[*-*->] 

Taking b equal to we have F = x - -i = x _ 

2 3 27 

and G = Therefore, * being = x, 

a 3 — * 5 ]' Xg'+ 3 gx % + 3^ 6 * 

Q 

and q (the value of _j— when x is = b) being a negative 
.quantity, y is a maximum when x is = 

27 

Taking b equal to a , we have F = * —, a Y~ 1 = x~^I \ m 3 

snd G "* ^^0\r Therefore " is = ~ f» which 

being contrary to our fuppofition, it follows, that v is neither 
a maximum nor a minimum when *• is = a. 


Example VI. Suffqfe y = a — jHHL* + „ 

, a* a 

is invariable and pofitive. 

Then will 

tw] be = lJH 3!- iiZEI = *s^x*-^x~ 

”* 8 ^Kj. a -r 7 ii+ 4 n 

where ,v is equal to or to a . 

Taking £ equal to wc have F .= x ~ = a: • — 

and 


9 6 


THE RESIDUAL 


and G = 


Therefore, m being = i, 


2a x x 5 • a 


and q (the value of - when a: is = b) being a negative 
quantity, y is a maximum when x Is = 

Taking b equal to <7, we have F = x — a — x~— i™, and 
G = - y .^ r=. Here m, which is manifeftly = I, 

is agreeable to our fuppofition ; but, the value of G being 
imaginary when x is greater than a , it appears, (as it likewife 
does from the equation of the curve,) that, when x is equal to 
a, the curve is not continued on both fides of the ordinate ; 
therefore y is not then a maximum or minimum , within the 
meaning of our explication in Art. 1. and 2. 

It may be worth while to enquire concerning the point of 
the curve to which the ordinate d (determined as above) corre- 
fponds, when m comes out contrary to our fuppofition, or G is 
not real, both when x is greater and when lefs than b: and that 
I purpofe to do, in the next article. 


Fig. 38. We have found, in Chap. 5. that —-— is the general value 
39- O-f -y] 

40. of the fubtangent ; therefore [x_^y] is the quotient of the 

4 1 - ordinate divided by the fubtangent. Now it is obvious, that the 
^ faid quotient will vanifli, when, the ordinate being finite, the 

44 - tangent is parallel to the bafe ; and that ^ the reciprocal of 

the fame quotient will vanifh when the tangent coincides with 
the ordinate. When the tangent is parallel to the bafe, the 
49. ordinate may be a maximum or minimum , as in Fig. 38, 39. or 
it may pafs through a point of contrary flexion, as in Fig. 40. 
or correfpond to a cufpid, as in Fig. 41, 42* 43 * when 

the tangent coincides with the ordinate, fuch ordinate may be a 
maximum or minimum , as in Fig. 44* 45 * or ^ ma y meet the 
curve in a point of contrary flexure, as in Fig. 46. or correfpond 

to 


97 


Chap. VIII. ANALYSIS. 


to a cufpid, as in Fig. 47, 48. or touch a continued arch, as 
in Fig. 49. It appears therefore, that, b being a value of x in 

the equation [x y] — 0 , or —-— “ 0 , and d the correfpon- 
[*—}'] 

dent value of y, (as fuppofed above,) the ordinate d may be a 
maximum or minimum , or correfpond to a point of contrary 
flexion or reflexion, or touch a continued arch : moreover, the 
tangent to the correfpondent point of the curve will be parallel 
to the bafe, or coincide with the ordinate, according as b is de¬ 
termined from the equation [xj-y] = 0, or — l -— — 0 j i. e. 

I 

according as m, in the equation y — d = x — b\ n ^ 1 x is 
pofitive or negative. 


If the ordinate d correfponds to a point of contrary flexure, 

x — ^1 x Q> the value of y — d } muft be negative when 
* is lefs than b , and pofitive when x is greater than b ; or 
negative when * is greater than b , and pofitive when * is lefs 

than b. Therefore x — b\ muft, in that cafe, be negative or 
pofitive, according as x is lefs or greater than b j and confcquently 
m -\-i mull: be an odd number, or a fraction whofe numerator 
and denominator are both odd numbers. From whence it fol¬ 
lows, that ?n mull: then be an even number, or a fraction whofe 
numerator is an even number and denominator an odd number. 
It is evident therefore, that, when m is as juft now fpecified, 
and G is real both when x is greater and when lefs than the 
ordinate d meets the curve in a point of contrary flexure; and the 
tangent at that point will be parallel to the bafe, or coincide with 
the ordinate, according as m is pofitive or negative : moreover, if 
x be increafed after being equal to b, y will, at the fame time, 
increafe or decreafe, according as q is pofitive or negative. 


The celebrated Marquis De L’Hospital diftinguifies cufpids (or 
points of reflexion,) into two kinds: when the branches of the curve 
which form the cufpid have their convexity towards each other , the 
ctijpid is faid to be of the firft kind ; but of the fecond kind, when 
the convexity of one of thofe branches is towards the concavity of the 
otherr— In what follows, we will obferve the fame diftinftion. 


O 


Let 


9« THE RESIDUAL 

Fig. 50. Let AE be the bafe upon which the abfcifla x is meafured 
when y is confidered as the ordinate, and let AF be drawn pa- 
* rail el to the faid ordinate. Then, y being fuppofed to denote any 

abfcifla AD, meafured on AF ; and x the correfponding ordinate 

DP ; we have, by the preceding article, [y — a] = 0, cr - 1 
1 . t f y 

— o, when x is a maximum or minimum. Therefore, [x 

(by Art. 9. Chap. 2.) being = —I— , —L_ w m be = 0, or 

r i ■*] [*—yl 

1 * —Xl = °> when x is as juft now mentioned. Moreover, 

from the equation = * — b\ m x Q, we get =r 

_ — 1 y 

y — 7 f +1 x "■ Therefore, by the lafl article, -— mufl 

^ Q»+i m 1 

be an odd number, or a fradion whofe numerator and denomi¬ 
nator are both odd numbers, when a: is a maximum or minimum: 
Hence it follows, that m mull then be a fradion whole numerator 
is an odd number, and denominator an even number.—Now, 
when DP (— x) is a maximum or minimum , CP (parallel and 
equal to AD = y) either touches a continued arch; or corre¬ 
sponds to a cufpid of the firfl kind, with the tangent at the point 
of reflexion parallel to the bafe.—It is manifefl therefore, that, 
if, when DP is equal to b , the curve be continued on both fidcs 
of that line, m mud be as juft now fpecified j and the ordinate 
d will correfpond to fuch a cufpid as we lafl mentioned, or touch 
a continued arch, according as m is pofitive or negative. 

From what has been laid it is evident, that, m being any number 
or fradion whatever, if the curve be not continued on both fides 
of the ordinate d 3 that ordinate will correfpond to a cufpid of the 
5 exce P l when m is a fraction whofe numerator is an 
odd number, and denominator an even number, in which cafe 
the faid ordinate may correfpond to a cufpid of either kind, or 
touch a continued arch. 

Corollary I. The value of [xj-y\ being exprefTed by a 
fradion, if, by fuppofing the numerator thereof = o y x be found 
equal to (any quantity) b ; and, by fuppofing the denominator 
^ °> x be likewife found equal to b it may, in fuch cafe, 

happen, 


99 


Chap. VIII. ANALYSIS. 

happen, that m (hall be = o. When it fo happens, the ordinate 
correfponding to the abfcifla b will meet the curve in a double 
or triple &c. point, according as q has two or three &c. values. 
If, m being = 0, q has two equal values, the faid ordinate will 
correfpond to a point of the curve where two branches thereof 
touch each other, either forming a cufpid there, or being con¬ 
tinued on both fides of fuch ordinate : and, if the values of q are 
all imaginary, fuch ordinate will correfpond to a conjugate point. 

The pofition of the tangents to the feveral branches of the 
curve, at fuch double or triple &c. point, will be known by 
means of the feveral values of q : for, m being = 0, q is the 
value of [x-^y] when x is equal to b ; and therefore q will then 
be to unity, as the ordinate to the fubtangent. 

Corollary II. m being a politive odd number, or a pofitive 
fraction whofe numerator and denominator are both odd num¬ 
bers, as many (ingle different real values as q has, fo many dif¬ 
ferent branches of the curve are continued on both (ides of the 
ordinate d, touching each other in P, the point to which the faid 
ordinate corrcfponds : and two equal values of q denote two con¬ 
tinued arches of equal curvature at P, and both -turned one way; 
or a cufpid of the fecond kind, at that point. 

The tangent, at P, to fuch cufpid, or continued arches, will 
be parallel to the bafe; and, according as q is negative or politive, 
the branch to which it relates will, at P, have its convexity up¬ 
wards or downwards. 

Imaginary values of q indicate a conjugate point at P , m being 
as in this, or the next Corollary. 

Corollary HI. m being a negative odd number, or a ne¬ 
gative fraction whole numerator and denominator are both odd 
numbers, as many fmgle different real values as q has, fo many 
different branches of the curve are continued on both (ides of 
the ordinate d, each forming a cufpid of the firft kind, at P : and 
two equal values of q denote two fuch cufpids, both pointing one 
way 5 or a cufpid of the fecond kind, at that point. 

The tangent to each of thefe cufpids will coincide with the or¬ 
dinate ; and, according as q is negative or pofitive, the cufpid to 
which it relates will point upwards or downwards. 

O 2 CoROL- 


100 


b 


THE RESIDUAL 

Corollary IV. m being an even number, or a fra&ion 
whofe numerator is even and denominator .odd, as many fingle 
different real values as y has, fo many different branches of the 
curve are continued on both tides of the ordinate d , each having a 
point of contrary flexure at P : and two equal values of q denote 
two fuch branches, both turned alike ; or a cufpid of the fecond 
kind, at that point. 

The tangent to fuch cufpid, or point of contrary flexure, will 
be parallel to the bafe, or coincident with the ordinate, according 
as m is pofltive or negative; and, according as q is pofitive or 
negative, y will be greater or lefs than d, when * is taken greater 
than b . 


Corollary V. m being a pofitive fra< 5 tion whofe numerator 
is an odd number and denominator an even number, as many 

fingle different pofitive real values as q and — i T x q have, fo 
many cufpids of the firffc kind will be formed at P : and two 

equal pofitive real values of y, or of — i \ m x y, denote two fuch 
cufpids, at that point, both turned one way; or a cufpid of the 
fecond kind there. 

The tangents to fuch cufpids will be parallel to the bafe ; and, 
(x being fuppofed pofitive when the abfciffa is on the right of 
the point where it begins,) according as the value of q or 

— J 1 x y is real, the branch or branches forming the cufpid, 
to which fuch value relates, will be on the right or left of the 
ordinate d. 

in being as in this, or the following Corollary, a conjugate point 
is indicated, if neither q nor — i \ m x q be real. 


Corollary VI. m being a negative fraction whofe numera¬ 
tor is odd and denominator even, as many fingle different pofitive 

real values as q and 1] x q have, fo many continued arches 
will be touched by the ordinate d: and two equal pofitive real 


values of y, or of — i \ n x y, denote, that d is a tangent to a 
cufpid of the fecond kind, at P ; or that d touches two con¬ 
tinued arches of equal curvature at that point, and both turned 


one way. 


Moreover, 


101 


Chap. VIII. ANALYSIS. 

Moreover, (x being fuppofed pofitive when the abfcifTa is on 
the right of the point where it begins,) according as the value 

of q or— i\ m x q is real, the continued arch, or the branches 
forming the cufpid, to which fuch value relates, will be on the 
right or left of the faid ordinate d. 

Scholium, m being any number or fra< 5 tion whatever, if q 
has two equal values, P, indead of being as above fpecified, 
will fometimes be a conjugate point. And, from this oblervation 
and what is faid above, the confequence is obvious, when, m be¬ 
ing of any value whatever, q has three or more equal values. 

Thefe Corollaries are of great ufe, for preventing midakes 
concerning the greated and lead: ordinates, and for afcertaining 
the true form of a curve from the equation thereof; as will 
farther appear by the examples in the following articles. 

5 - 

If the value of y be not expreffed in terms of x, (as is fre¬ 
quently the cafe,) the value of [x^y] will be exprefled in terms 
containing both x and y. In which cafe, m and q will not be 
found as when [x^y] is a function of x without y being con¬ 
cerned therein. But they may then be found as follows. 

Suppofing b to be a value of x, and d the correfpondent value 
of y, when either the numerator or denominator of the value of 

[x _s_ y] is = o ; fubditute d x — 7 f + I x Q_ for v, in the 
equation of the curve j (the terms being all on one fide, and 
confequently = o ;) and, in the expreflion which refults upon 
fuch fubditution, take m of fuch a value, (greater than — i,) 
that the faid expreflion being divided by the lowed power of 
x — b in the feveral members thereof, the quotient fhall not, 
upon taking x equal to b t confid of lefs than two members, all 
the terms in which Q^is not concerned being accounted but one 
member. Then, fuch quotient being = o, by writing therein 
q indead of Q, and the value of b indead of x, q will from 
thence be determined. 

m and q may alfo be found by fubdituting d -f- x — b\ x q 
fory in the value of fx _i. y}. For the refulting exprelfion being 

divided 


102 


the residual 

divided by m +1 x x — !\ m , the quotient, when * is therein 
taken equal to b, will, by what is proved above, be equal to q ■ 
i. e. it R be put for the expreffion which is obtained by fubiti- 
tuting, as^juit: now mentioned, in the value of [a; j-y] t 

x mi w ^ en x will be ==/»-{- i x y : 

from whence m and q may be readily determined, by taking- m 
ot fuch a value, that the numerator and denominator of the value 

of the expreffion (hall each be divifible by one and the 

fame power of x — b ; and, when each is divided by fuch 
power, neither of the quotients (hall vanilh upon taking x equal 

Remark. The exponents of the feveral powers of * and y 
being pofitive integers, the conclufion will be exadtly the fame, 
(and the procefs more concife,) if, when the numerator of the 
value of [x-uy] vamlhes, and the denominator, at the fame time 
does not vanilh, we write b for a- in the denominator, and only 
dioxy in both numerator and denominator: Alfo, if, when the 
laid denominator vamlhes, and the numerator, at the fame time, 
does not vanilh, we firft write b for * in both numerator and 
denominator, and then fubftitute d + q f or v ; n thr 

thtZr^’r u nly a d f ° r y in the numerat °r i omitting (in 
this htter cafe) in the denominator, every term wherein y is not 

concerned before fubftitution, and every term but thofe wherein 
the lowed; power of q is concerned after fubftitution. 

For the terms omitted in the value of R in confequence of thefe 
rules would, if they were therein written, always vanilh in carry¬ 
ing on the procefs ; as will ealily appear, upon confidering what 
we juft now faid concerning the expreffion - R . 

Example I. Let the equation of the curve be x*— axy‘ 4- 
cy — o, a and c being invariable pofitive quantities. 

Then, by refidual divilion, we have 4*’— ay' — 2 axy[xu-yl 
* 3 c y {* — y] = o ; and confequently 

2a xy — 


Suppofing 


Chap. VIII. ANALYSIS. 

Suppofing the numerator 4*' — ay 1 = 0, we, from thence and 

the equation of the curve, find at = 0, alfo x = ££; and the 

4V 

refpeftive values of y are 0, and . 

Suppofmg the denominator 2axy — 3 cy x -=z 0, we, from hence 
and the given equation, find * = 0, alfo * = ; and the re- 

fpedlive values of y are 0, and 


Taking b = 0, d will be = 0, and d -f- x m — 

(the quantity to be fubftituted for y , in the equation of the 
curve,) = Qx m + \ Therefore * 4 — ax 2m + 3 Q 1 + cx 3m+3 Ql 
will be = 0 j where, it is obvious, that, to determine q as taught 
above, m may be = 0, alfo m — 

• = f> * 4 — ^f’Ql'will be = 0. Therefore, 

in this cafe, (dividing by.v } , and then taking x = 0,) — 

is = 0, or — ^q = 0. Here y has three values, two of 
which are each = 0, and the third = 

c 

Therefore, by Cor. 1. of the lafl: Art. the point (A) where Fig. 51. 
x begins is a triple point in the curves and two of its branches 
touch each other at that point. Moreover, q being to unity, as 
the ordinate to the fubtangent j and the value of q which relates 
to the faid two branches being — o, the tangent to thefe branches 
at A coincides with the bafe AF: and, the value of q which 

relates to the third branch being —, the tangent (AG) to this 

branch at A makes an angle GAH with the bafe, fuch, that GH 
(which is fuppofed parallel to the ordinates of the curve) is 
to AH, as — to 1. 

c 

If n\ be equal to cxl will be = 0. 


Hence 


104- 


* 


THE RESIDUAL 

Hence i — aq' = o, and q = -A r . Therefore, by Cor. 5. 

of the laft Article, there is a cufpid of the firft kind at A, the 
tangent to which is coincident with the bafe; and (* being 
always fuppofed pofitive when the abfcilfa is on the right of the 
point at which it begins,) the branches forming fuch cufpid arc 
on the right of the fame point: which agrees with what we 
difeovered by confidering m equal to 0. 


2dly. Taking b equal to d will be = 2 ^-*. Therefore 

R ^ L ^ L 

r==p (abridged agreeable to our Remark) is = — X 

__ t+ _ A 

4 9(6 _ 


x — bXx 1L +bx + b ' 1 


labd — yd z A . ^ X 2 abd — yd z * 2 abd — yd z 

= m -f- i X q when x is = b. Whence it is manifeft, that 


mis — i, and q r= . Therefore, q being a negative 

quantity, it follows from what is faid above, that the ordinate 
■BP C~ d) is a maximum when (AB) the correfponding abfcilfa 
is equal to b. 


3 dly. Taking b equal to d will be = r —. Therefore 
R 3 £ 3 c 

(abridged according to our Remark) is = *— b\~ m x 


4^ 3 

2 ab — bed 



= m + 1 x q when x is = 


b. 


It appears 


therefore, that m is = —. - 1 , and * b ■ ad -— = ~q ; Hence 

_a zab — bed X g 7 

q is found = J ad " . 

Now, 4 b' being greater than ad\ and ab lefs than 3 cd, the 

value of q is imaginary, and the value of — 1 \ m x q is real. 
Therefore, by Cor. 6. of the laft Article, the ordinate ef (— d) 
touches a continued arch when (Ae) the correfponding abfcilfa is 
equal to b j and the arch fo touched is on the left of the faid 
ordinate. 


Example 


Chap. VIII. ANALYSIS. 

Example II. Let the equation of the curve be x* —* 2x l y •— 
4xy‘ + y x — f = o. 

Then, by refidual divifion, we have 4*’ — 4*y — 4/ —. 
2x [x y ] — 8 xy [x ^.y] + 2 \y [x y ] —. 3/ [x jL.y] = 0 ; and 
confequently f-v -i- yl = - 4 ** 4:9 ^ 

Suppofing the numerator 4*' _ 4 *y _ 4Jf * = £)( we> f rom 
thence and the equation of the curve, find * = o, alfo * = 2 r 
and the rcipective values of y are o, and —• 4; y being, in fuch 


cafe. =-. 

’ I — X 4” X x 

Suppofing the denominator 2x s + 8 xy — zy + 3 y z — 0, we, 
from hence and the given equation, find x = o, or x = 2, or 

A * ^ > and the relpecftive values of^y are o, — 4, and ^ y 

being, in this cafe, = %*' — %x* 

i 6 at — 20 a; 1 — 2 * 

lft. Taking b equal to 0, </willbe = *, and d + 

* ^7 9 ^ • # Therefore, this lafi quantity being fubfiituted 

lnitead of y in the equation of the curve, we have 

at* — 2Q/’ +3 — 4 Q\- 2m+3 + (£x 2m + 2 - Q^+3 _ Q . 

And here, to determine q, m muft be = i. 

TO W CqUCn ^ 1 2Q ~=~ 40 ^ + QL— QV is = e ; from 

whence we have i - 2? + q‘ = 0 , And /having two equal 
values, each == i, there is, agreeable to Corol. 2. of the lafi: 
Article, a cufpid of the fccond kind, at the point A, * where Fig. 

x be- 


i.-*, B ^ C ° rol,a . r y of the laft Article, there is either a cufpid of the fecond 

To know 1 Jh?rh nt ^ ; ° r the u afe touches two continued arches, at that point. 

«nccX ,t h v ,i h ' f CU o VC h " **«. or nat ' farther enquiry 

concerning the value of Qjs requifite.-Having found, that = i when i 

is = o, we may fuppofe Q^= I +Ar"<l, n being fome pofitive number or fiac* 

tion, and (^foch a funfiion of at, that neither it nor its reciprocal fhall ranlfli 
upon taking x equal to o. 

Therefore i 4. jA^be fubfiituted Lnftead of Qjn the equation 1 — 2<J^ 
* ^ 4x.- ) c QjV = 0: And then, from the refulting equation, n may 

P he 


THE RESIDUAL 

* begins; fhe tangent to which coincides with the bafe, and the 
convexity of the branches forming fuch cufpid is downwards. 

2dly. Taking b equal to 2, d will be = — 4 > and x 4 + 8x' 
— 6 4 x+ 80 — 2 xx — 16X + 28 x x— • 2l m+ ' x Q_+ 

13 — 4AT X X — 2 1 x — x — 2 \ _X 0 - It 

appears therefore, that ** + 4* + 20 x x — 2I + 2 8 2 x 
X r=l \ m+2 x Q_+ 13 — 4*x* — 2| X o. — 

_21 3m + 3 x QMs = a; and that, to determine q, m muft be 

= 0. Confequently (dividing by x — 2 1 , and afterwards tak¬ 
ing * = 2, and Q_= ?.) 3 2 + 2 4 ? + 5 ?‘ is = °> and 

a — _ 11 + iv/HTl. Here, thefe values of q being ima- 

* 5 — 5 

ginary, the ordinate d , by Cor. i. of the laft Art. correfponds to 
a conjugate point, when the abfcifla is = 2. 

3 dly. Taking b equal to ±, d will be = £ Therefore 

■ ^ ^ (abridged agreeable to our Remark) is := x — /i 

x ■— A™ 

.. 4* 3 —-4^— 4^ » 1 x ^ w hen a: is = It is evident 

ir»i +1 1 * 


[O -7> 

-q , x — b\ 

in* 


r 1 , 4* 3 ~ 4^ — 4^* _ 1 

therefore, that « is = — and ,0 — # 

. _* 7 ? 

Hence q is found = ^ x b 1 — ^ ^ • 

be determined, and alfo J, the value of Q.when ^ is = O, in the fame manner 
as m and q are determined above. 

By proceeding in that manner, rt is found = and y = + 2 * Confe¬ 
quently y (= Q* 1 ) is = **CL» C>_ being fuch a function of *, that its 

value is ± 2 when * is = 0 . Whence (Aj. being imaginary when a- is ne¬ 
gative,) it appears, that the curve is not continued on the left of the point A. 
Therefore, by what is faid above, there muft be a cufpid of the fecond kind, a 

A, A 2 nd m by i pu,fain 6 the fame method, we may be fatUfied in other fuch ambi- 
guous cafes. Confe- 


— -q : 

2 * 


Chap. VIII. ANALYSIS. I07 

Confequently, the value of q being imaginary, the value 

of — i\ m x q is real: and, by Corol. 6. of the laft Art. the 
ordinate BD (= d) touches a continued arch, when (AB) the 
correfpondent abfcifla is equal to b 3 and the arch fo touched is 
on the left of the faid ordinate. 

Example III. Suppofe x T —. jxy* -f y s = o. 

Then will [x «i_y] be = gl-T-X. 

6 xy — 

Suppofing the numerator 3X* — 3 y x = 0, we, from thence 
and the other equation, find x = 0, or x = 2, or x = — 2; 

and the refpeftive values of y are 0, \/ 2, and — V 2. 

Suppofing the denominator 6 xy — £y + = 0 , we, from hence 

and the firft equation, find x = 0, or x = ~ x x/^, or 

5 * 

* = — ~ x V 2, and the refpeftive values of y are 0, 

~ x V 2, and — “x/2. 

5+ 5* _ 

ift. Taking b equal to 0, d will be = 0, and d -f* x — b\ mJrl 

x Qj= Qx u \ Therefore x J — 3 + QL* 5 i« 

= 0; where, it is obvious, m may be = 0, alfo m — - 

If be equal to 0, 1 — 3 q v will be = 0 .* Hence q is found 
, or q = — . By thefe two values of q it appears, 

that two branches of the curve interfedl each other at A, the Fig. 53. 
point wherex begins: and that the tangents (AG, Ag) to thofe 
branches, at A, make equal angles (GAH, gAh) with the bale 
AE 5 and fuch that GH and gb (which are fuppofed parallel to 
the ordinates of the curve) are to AH and Ah reipe&ively, 


If m be equal to — — 3^* + q' will be = 0: Hence 

we have q = 3V Therefore, by Corollary 4. of the laft Article, 

P 2 A 


THE RESIDUAL 

A is a point of contrary flexure, in another branch of the curve 5 
the tangent to which, at that point, is parallel to the ordinates; 
and, q being pofitive, the value of y correfponding to this branch 
will be pofitive when x is pofitive. 

2dly. Taking b equal to Hh y/ 2 , d will be = + v/2. 
■“ 

Therefore rrr— (abridged according to our Remark) is = 


r. s/ 2 X X + \ 


z= 7 n-\~ixq 


- -Ml 6 Id — 5 d* - bbd — r * 

when x is = b. Whence it is evident, that m is = 1, and 
q = ^ b - ^ . Therefore, q being negative when b is = v/ 2 

and pofitive when b is = — \/ 2, it follows, from what is faid 
above, that the ordinate BP (= \/ 2) is a maximum when (AB) 
the correfpondent abfcifla is = v/ 2 : and that the ordinate bp 
(= s/ 2) is a minimum when (Ab) the abfcifla correfponding 

thereto is equal to %/ 2 ; b being on the contrary fide of A, 
from B j and the point p below the bafe. 

3dly. Taking b equal to + \ X 2, d will be = + x y/ 2. 

Therefore — » (abridged agreeable to our Remark) is = 

AT- bV ^ ^ 


— 15 tPq . x — !a 


— = m -[- 1 x q when x is 


Whence it is manifeft, that m is = — and q = . 

Confequently, the value of q being imaginary or real, and the 
value of — iV”x q real or imaginary, according as b is taken 

equal to ~ x v/ 2 or — x V 2; the ordinate EF (by Cor. 6. 
5 T 5 t 

of the laft Art.) touches a continued arch, when (AE) the 
ablcifla anfwering thereto is equal to \ x >/ 2, and the arch fo 

touched is on the left of the faid ordinate : likewife the ordinate 
*J touches a continued arch, when (A?) the correfpondent abfcifla 

is 


109 


Chap. VIII. ANALYSIS. 

is equal to AE; e being on the contrary fide of A, from E ; 
and the point f below the bafe; and the arch fo touched is on 
the right of the faid ordinate ef 

6 . 

\Vhen m is = 0, and the curve is continued on both fides 
of the ordinate d t that ordinate may pafs through a point of 
contrary flexure, the tangent to which is oblique to the bafe and 
ordinate.—Suppofe P to be fuch a point in the curve Vf and let Fig. 54. 
P# be a tangent at that point. Then, the ordinate ef interfeft- 55 * 
ing that tangent in g, being called y ; the ordinate BP, d ; and 
the abfciflas AB and Ae , b and x refpe&ively; will be 

= y — d — x — b x q, q denoting the value of [x _i_y] when 
x is = b, i. e. q is = d -7- Subtang. Which value of ef — eg , 
it is manifeft, will (as when it relates to Fig. 54.) be pofitive 
when x is greater than b , and negative when x is lefs than b ; or 
(as when it relates to Fig. 55.) pofitive when x is lefs than b, 
and negative when x is greater than b. Therefore, by Chap. 4. 

Art. 3. x — b\ n Q ^may be aflumed — y — d — x — b x q\ n 
being an odd number, or a fraction whofe numerator and deno¬ 
minator are both odd numbers; and Q^fuch a function of x , 
that neither it nor its reciprocal Ihall vanifh upon taking x equal 

to b. From which aflumed equation we have yr=zd-\-x — b .q 
— b\\ Therefore, y being before fuppofed = d -j- 

a: — b . Q, it is evident q x — b\' 1 x Q^will be r= Q^; 

where, it is plain, n muft be greater than i». 

Let therefore q -f x b^^'x QJ^e fubftituted inftead of Q, 
in any equation found by fubftituting, and taking m equal to 0, 

as in the preceding article : And then, fuppofing q to be the 

value of Qjvhen x is = b , n and q will be determined in the 
fame manner as are m and q in that article. 

By the value of n (fo determined) it will appear whether the 
ordinate d correfponds to a point of contrary flexure, or not; 

and, 


no 


THE RESIDUAL 

and, if it does correfpond to fuch a point, the value of q will 
fhew the tendency of the curve on both Tides thereof, the con¬ 
vexity of the branch on the right of that ordinate being down¬ 
wards or upwards, according as q is pofitive or negative. 


Example. In Examp. 3. of the lajl article , y is = o, when 
x is = o : and by fubftituting Qx^ + 1 inftead of ^ in the equa¬ 
tion of the curve, we get x' — 3Qjc 2ot + 3-[- Q^f 5 OT +S== 0 > 
from whence, by taking m equal to 0 , and dividing by x\ we 
have 1 — 3QI+ Q V = o. Hence q (the value of Q^when * 

is =^o) is found = + ->/•“• 

Subftituting q -f x QJnftead of Q^, we have 
1 — 3 9 % — 

+ ?V + J?V+'QJ- ic ? V”Ql+ ioq'x3 n 'Ql+j 

“ 2 Ql-f 

Here, it is obvious, n muft be = 3 ; and confequently 
f — 6 qq = 0. 

Therefore q is = (= —, as well when q is = — J ^ 


as when q is: 


v 5+’ .” 3 

and it appears that the two branches of 


Fig. 53. the curve, whofe tangents at the point A are AG and Ag } have 
each a point of contrary flexure at A, as in Fig. 53. 


7 - 

F; g . e+ , P being a point of contrary flexure, and the abfcifla and 
5S-. ordinate correfponding thereto being called 0 and S refpeitively; 
and the value of [x — y], when x is = ( 3 , being denoted by q j 

v — S — x— 0 . q, as is (hewn in the preceding article, will 
be pofitive when * is greater than j 3 , and negative when a- is lefs 
than (3 j or pofitive when .v is lefs than 0 , and negative when -* 


Ill 


Chap. VIII. ANALYSIS. 

is greater than 13 , how near foever x be taken to There¬ 
fore, y — x .~ z JLJ being pofitive or negative according as 

y — $ — * — Q.q is pofitive or negative, it follows from Art. u 
Chap. 4. that the value of y ~~ ^ or its reciprocal, 

will be = 0, when x is = ( 3 . _ 

Now, by what is taught in Chap. 2. the value of -~~~ - ^^ r- ‘ q , 

when * is = jG, (i. e. when y is = 3 ,) is = the value of 


Confequently [a: j±. y] is = 0, or ■ 


= 


when the 


ordinate y correfponds to a point of contrary flexure : And the 
fame conclufion follows, from confidering that [x _i_ y], the 
quotient of the ordinate divided by the fubtangent, is then a 
?ninimum , as in Fig. 54. or a maximum , as in Fig. 55. 


Moreover, fuppoflng /3 to be a value of x, and <£ the corre- 
fpondent value of y, in either of the equations [x ^~y] = o, 

or —-— = 0 ; andafliiming x — x Qj= -— ^1 

[x ^ y ] * — & 


being fuch a function of x, that neither q , the value of Q^ 
when x is = ( 3 , nor the reciprocal of that value (hall vanifh; 

we have y - ^~ * * 0 * * = Ci; from whence, when x 19- 

x — p\ n+z 


~ Q C — _ q. And 

* » + 2.x — 71 ” + 1 n + 1 . n + 2 . x — &\ n 

it follows, from what is before faid, in this and the 4th Chap, 
that, the curve being continued on both (ides of the ordinate J 1 , 
if n be an odd number, or a fraction whofe numerator and 
denominator are both odd numbers, [x^.y] (when x is =/ 3 ) 

will be a minimum or a maximum , according as q is pofitive or 
negative ; and the faid ordinate ^ fhall pafs through a point of 
contrary flexure. But if n be an even number, or a fraction 
whofe numerator is even and denominator odd, fuch ordinate 
will correfpond to a point of the curve where the curvature is 

nothing 


112 


THE RESIDUAL 

nothing or infinite , and the concavity of the curve is turned the 
fame way on both fides of that point. 

When [x JLy] is a fundlion of x without y being concerned 

therein, n and q will be determined in like manner as are m 
and q when [# y ] is fuch a function of x» 

To find n and q when the value of [x ^~y] confifts of terms 

in which both x and y are concerned, let <? + * —0 .? + 

x — i 3 ]” 2 x Q^be fubft ituted for y , in the equation of the 

curve; or x — fi . q x f 3\ n + 2 x q for y in either of 

the eauations it $ ^ — = a when x is = Q. or 

u „ + 2 . + x * 

- i XmiSm y 1 - - = q when x is = ( 3 . From either of 

n -f. i . n -J- a . x — bV 

which equations, after fubfiitution, n and q may be determined 
in the fame manner as m and q are determined above. 

Corollary I. n being an odd number, or a fraction whofe 
numerator and denominator arc both odd numbers, as many 

fingle different real values as q has, fo many different branches 
of the curve are continued on both fides of the ordinate <£, each 
having a point of contrary flexure at P, the point to which the 

faid ordinate correfponds: And two equal values of q denote two 
luch branches, both turned alike; or a cufpid of the fecond kind, 
at that point. 

Moreover, (x being fuppofed pofitive when the abfeiffa is on 
the right of the point where it begins) the branch of the curve 
on the right of the ordinate J* will have its convexity downwards 

or upwards, according as q is pofitive or negative. 


Corollarv 


Chap. VIH. ANALYSIS. 

Corollary II. n being an even number, or a fra&ion whofe 
numerator is even and denominator odd, as many (ingle different 

real values as q has, fo many different branches of the curve are 
continued on both f.des of the ordinate touching each other 
in P, and each having its curvature nothing or infinite , at that 
point, according as n is pofitive or negative: And two equal 

values of q denote two fuch continued arches, both turned one 
way ; or a cufpid of the fecond kind, at P. 

Moreover, according as q is pofitive or negative, the branch 
to which it relates will, at P, have its convexity downwards or 
upwards. 


Coroll ar y III. n being a fra&ion whofe numerator is odd and 
denominator even, as many (ingle different pofitive real values as q 

, ■ - I n it 

and 11 x q have, fo many cufpids of the firfl kind will be 

formed at P : and two equal pofitive real values of q t or of 

— ii* x q , denote two fuch cufpids, at that point, both point¬ 
ing one way ; or a cufpid of the fecond kind there* 


Moreover, (x being fuppofed pofitive when the abfcifla is on 
the right of the point where it begins,) according as the value 

of q or— 1 1 x q is real, the branch or branches forming the 
cufpid to which fuch value relates will be on the right or left of 
the ordinate 


Corollary IV. n being = o, as many (ingle different real 

values as q has, fo many different branches of the curve are 
continued on both (ides of the ordinate touching each other 

in P: and two equal values of q denote two continued arches of 
equal curvature at P, and both turned one way; or a cufpid of 
the fecond kind, at that point* 


”■3 


The 


it* THE RESIDUAL 

The tangent, at P, to fuch cufpid, or continued arches, will 

be oblique to the bafe and ordinate ; and, according as q is pofi- 
tive or negative, the branch to which it relates will, at P, 
have its convexity downwards or upwards. 

Having faid fo much concerning curves referred to a bafe, 1 think 
it unnecejjary to add any thing here relating to Spirals ;—to uhich 
ki?id of curves the intelligent Reader will readily apply the above 
method of reafonifig , 



THE 


Chap. IX. 


ANALYSIS. 


”5 


% » .3 1 *5£~tfat/'a.S'.*l£~M V-2f it a P 


* 


an 


tffTO 


* 




THE 

RESIDUAL ANALYSIS. 


CHAP. IX. 

Of the Asymptotes of curve Lines. 

a knowledge °f the tendency of the infinite branches 
gj* A ^-g of a Curve, when fuch there are, is requifite for the 
obtaining a clear idea of the figure of fuch Curve, 
>@0§0e0s09( and may fometimes facilitate the bufinefs of finding 
the moft diftinguifhed points therein; I fhall there¬ 
fore point out an eafy method of finding the Afymptotes to fuch 
branches, by which means their tendency may be very readily 
difcovcred. 

A branch of a curve when infinitely continued may be con- 
fidered as coinciding with its afymptote. If therefore m and n 
be refpeftively put for the fines of the angles which a rectilinear 
afymptote makes with any ordinate and the bafe; or for the fines 
of the angles which the tangent to a curvilinear afymptote, when 
extended infinitely, makes with any ordinate and bafe ; m will 
be to n, as to ['u.i.y], when the curve is infinitely con¬ 

tinued. Confequently, if from the equation of the propofed 
curve, (which fuppofe clear of furds,) we, by refidual divifion, 
(<y — y being the divifor,) deduce a fecond equation, and therein 

write m and n infiead of [v #1 and [<y j-y] refpeCtively, we 

OL.2 (hall 


u6 THE RESIDUAL 

fliall obtain the equation of an afymptotic line, one dimenlion 
lower than the given equation, where m and n will be lome 
* invariable quantities, which will be determined in confequence of 

this procefs. And, the propofed curve being above the fecond 
prder of lines, we, from the equation of that firft alymptotic line, 
by the like refidual divifion, and again fubflituting m and n inftead 
of [v _£_ x] and [v refpettiveiy, may deduce the equation 
of a fecond afymptotic line, two dimenfions lower than the 
given equation. Like wife, the propofed curve being above the 
third order of lines, we, by our fatd divifion and fubftituting as 
before, may derive, from the equation of fuch fecond alymptotic 
line, the equation of a third afymptotic line, three dimenfions 
lower than the given equation. And fo, with great facility, we 
may proceed with any given equation, till we get equations of 
alymptotic lines of every dimenlion lower than the given equa¬ 
tion : alfo, from the lowed of thofe equations, (viz. that of one* 
dimenlion,) we may, by continuing the fame procefs,. obtain an 
equation containing no other unknown quantities but m and n; 

from whence —, or — will be determined j and then all that is- 

n tn 

requifite, in the feveral equations of the afymptotic lines before 
found, will be known ; and confequcntly the tendency of fuch 
lines, and of fuch branches of the propofed curve to which they 
are afymptotes. 

Scholium I. The polition of the ultimate tangent * to a 
parabola being not determinable, fuch tangent only coming, at 
an unlimited diftance, almofl: to a parallelifm with fome certain 
right line : when the afymptotic line (exprefled by any equation 
found as above) is of the parabolic kind, it will appear by fome 
equation in the above procefs, that m (or n) cannot be taken 
ab'olutely equal to its apparent value in the final equation in the 
laid procefs, without occafioning an abfurdity j and the impofii- 
bility of a re&ilinear alymptote will be evident. Therefore, in 
fuch cafe, we are to confider m (or n) only as approaching very 

near to fuch apparent value; and accordingly,, inftead of ~ (or —^ 
fubftitute its value obtained from fome equation before found. 

* By ultimate tangent , I mean the tangent to an infinitely diftant point of 
the curve. ScHO- 


Chap. IX. ANALYSIS. 

Scholium II. It may happen that the afymptotic equation 
of two (or more) dimenfions, found by our procefs, will ex- 
prefs two (or more) right lines, inftead of exprefling a curve. 
In which cafe, to find* the moft fimple hyperbolic afymptote 
of the branches to which;a rectilinear one fo found relates, fome 
farther enquiry.is requifite. Suppofe the equation of a re&ilinear 
afymptote (found by the above, procefs) to be y — Ax -{- B. 
Then, it is obvious; the branches to which fuch right line is an 

afymptote may be expreffed by the equation y — Ax'-J-B 
/ being fome negative number or fra&ion, and Q^fuch a func¬ 
tion of x, that neither ( q ) the limit to which it converges, when 
x (being a very large quantity) is taken greater and greater, 

nor the reciprocal of that limit, fhall be = o. And it is 

likewife- obvious, that; jr= Ax + B -f- qj* is the equation of 
the mofl limple curve which may be an afymptote to fuch 

branches. Therefore let Ax -f- B -f- )e fubftituted for^ in 
the given equation ; and take p of fuch a negative value in the 
refulting equation,- that the fame, after it is divided by the highefV 
power of x therein concerned, fhall not, upon fuppofing x in¬ 
finitely great, confift of lefs than two finite members: From 
whence q may be determined. 

Example I. Let the equation of the curve be y 4 — - ax 1 )’ 1 -f» 
bx = o^a and b being invariable pojitive quantities . 

Then, by proceeding as above mentioned, we find the 
ift afympt. equal; 4 ny' ~ + 3 W = o ; 

— amy % 

2d afympt. equal, hny 1 — 4 amnxy -J-» 3^0 V= o; 

— an x l 

3d afympt. equal. 4 ny ~ + btn' = o; 

alfo — arnn 
where — is. = 0, or.— 

- m nt 


= 0 i 

= + y/ a . 

Taking 


ll 7 


ixS THE RESIDUAL 

Taking^- equal to a z and — a* fuccefiively, it appears, 
* by the 3d afympt. equat. that the curve has two redilinear afymp- 

totes, whofe equations are 2a T y — 2 ax + b = o, and — za T y 

— 2 ax -f- b = o ; and, by the 2d afympt. equat. that the two 

hyperbolas,whofe equations are 50/ — j±a T xy — tfV + 3^ = o# 

and 50/ -j- 40**^ — **** + 3^ = °> are afymptotes of the 
propofed curve; alfo, by the id afympt. equat. that the curve 
has for afymptotes two lines of the third order, whofe equa- 

tions are 4 a % y* 2 axy' — 2a x x'y + 3 bx = o, and —-4 a x y — 

2<7Ay r -f- 2 a T x % y -f- 3 bx x = o. 

Taking — equal to o, we have, from the 3d afympt. equat. 

b = o: which is abfurd, (as rb is fuppofed of fome value,) and 
therefore, according to Scnol. 1. (hews that there is no redilinear 

afymptote parallel to the bafe; (the final equation = o only in¬ 
dicating, that the ultimate tangent of fome parabolic afymptote has 
nearly that diredion;) and that, for the (aid afymptotic equation to 

be a true one, — mud not be confidered as abfolutely = o, but 

m 

only as indefinitely fmall, fo that being multiplied by a very great 
quantity, the produd may be fomething confiderable (viz. = b). 
Therefore, from that confideration, retaining the terms wherein 

the root — is found, (after dividing by m\) as being mod con¬ 
fiderable ; and rejeding thofc in which the higher powers of 

— are concerned, as inconfiderable ; we have ^ = b , or 

~ . Which lad quantity being fubdituted for — in the 

m 1<*y ^ J 0 m 

fecond afymptotic equation, ader dividing by m and rejeding 

the terms wherein ~ is concerned, we get bx ay* == o. 
m* 0 

Confequently the conical parabola exprefied by this lad equation, 
is an afymptote to the propofed curve. Moreover, by fubditut- 

ing -L inftead of — in the i ft afympt, equat. it appears that a 
2aJ m line 


Chap. IX. ANALYSIS. 

line of the third order, whofe equation is by *— a z xy x -|- abx r = o, 
is an afymptote to the fame curve. 

Example II. Let the equation of the curve be 

xy' + V — — bk' — cx — </ = o: 

Which is the chief equation in Sir Isaac Newton’; Enumerate 
Linear . Tert. Ordinis. 

Then, by refidual divifion, we find the 
i ft afympt . equat. j^ nX y + en — 3 amx% — 2 bmx — cm == o j , 

2d afympt. equat. + — 3 am x x —- bm x = o: 

alfo mn l — am' = 0; 

where ~ is = o, and — = -f -L. 
n n — 

It is evident therefore, that, when a is a pofitive quantity, 
the curve has three rectilinear afymptotes one of which, as 
appears by the fecond afymptotic equation, is the principal’or¬ 
dinate, and the other two are exprefled by the equations 

2a 'y — 2a * — b — O, and 2ay -J-. 2 ax -f b = o. More¬ 
over, by taking equal to o, and — -4 fucceftively, in 

the 1 ft afympt. equat. it appears that the curve has for afymptotes 
the three hyperbolas, whofe equations are 2xy + e = o, 

y + 2a~xy + a'e — 3 ax 1 — 2 bx — c = o, andy*«— 2 a l xy 
— a * e — 3 ax% — 2 bx — c = o. 

When a is = o, mn is = o; whence — — o, or —. =r o. 

Therefore, if b be not alfo = o, it is manifeft, from theVecond 
afympt. equat. that the curve cannot have a rectilinear afymptote 

parallel to the bafe: But (rejecting as inconfiderable) we 

have, - 


Ijr 9 


120 


THE RESIDUAL 

have, from that afymptotic .equation, £- = -L. Which laft 
quantity being fubftituted for in the ift afympt. equat. 

after rejecting ^ as inconfiderable, we get y l — bx — c = o. 

Confequently the parabola exprefied by this laft equation is an 
afymptote of the propofed curve. 

If both a and b be equal to o, the 2d afympt. equat. entirely 
vanilhes when — is taken equal to o j and from the ift afympt. 
eqnat. we then have y % — £ — o t which is an equation to two 
right lines parallel to the bafe. From whence we have y ~ c T > 

and y = — r 1 . Therefore, in this cafe, the curve has two 
rectilinear afymptotes parallel to the bafe, one above and the 

other below; and the diftance of each from the bafe is f 7 . And 
it is plain, thefe afymptotes, when c is =3 o, both coincide with 
the bafe. 

It is obfervable, that, in every cafe, the principal ordinate is 
an afymptote. 




THE 


Chap. X. 


ANALYSIS. 


12 I 


| 0>»>B<>n<>H<>G«H<C<e 1 

THE 

RESIDUAL ANALYSIS. 


CHAP. X. 

Of the Diameters and Centers of curve Lines. 

ing the forms and properties of curves, we 
:ntly have occafion to enquire concerning 
. , diameters and centers: It may therefore be 
k wort ^ to (hew the ufe of our dodrine in 

hich enquiries. 

r. 

When any number of parallel right lines, drawn between two 
branches of a curve, and terminated by thofe branches, are all 
bilefled by fome line, I call fuch bifedting line a diameter of the 
ftrft kind . And, any number of parallel right lines being drawn 
between two extreme branches of a curve, and cutting the inter¬ 
mediate branches; if thofe parallels be interfered by fome line, 
(wnich is not fuch a diameter as juft now mentioned,) fo that the 
of the parts of any parallel, lying on one fide of that 
lnterlecting line, and terminated by it and a branch of the curve, 
be equal to the part, or aggregate of the parts, of the fame 
parallel, terminated in the fame manner, lying on the contrary 
^. e f ame interfering line; I then, for diitinrion fake, 
call fuch interfering line a diameter of the feeond kind . When 

R all 



122 


k 


Fig. 56, 


THE RESIDUAL 

all the re&ilinear diameters of this fecond kind, appertaining to 
any curve, interfed each other in one and the fame point; fuch 
point of interfedion, I prefume, may, not improperly, be called 
a diametric foie. 


2 . 

A point bifeding all right lines drawn through it, and ter¬ 
minated by two branches of a curve, I call a center of the Jirft 
kind. And, any number of right lines being drawn through a 
certain point, terminating at the extreme branches of a curve, 
and cutting the intermediate branches ; if the aggregate of the 
parts of any fuch line, lying on one fide of that point, and ter¬ 
minated by it and a branch of the curve, be equal to the part, or 
aggregate of the parts, of the fame line, terminated in the fame 
manner, lying on the contrary fide of the fame point y we may 
denominate fuch point a center of the fecond kind L 


3 - 

EFG being any curve, whofe abfcifla AB, is called x j and 
ordinate BF, y : fuppofe efg to be another line, whofe abfcifla 
Ab, and ordinate bj\ (parallel to BF,) are denoted by v and w 
refpedively : alfo fuppofe the right line F/, which we will call //, 
always to make invariable angles with thofe ordinates. Then, 
drawing fh parallel to AB, and calling the fines of the angles 
fh¥ , fFb, and Ffh ; k } m, , and n refpeaivelyj fh will be 

= and Fh = Confequently, writing 2 inftead of x 

will be = v + mz , and y = w + nz. Which values of x and 
y being refpedtively fubftituted for their equals, in the equation 
of the curve EFG, the nature of that curve will be exprefled by 
an equation (hewing the relation of the three quantities v, w , and 
z ; which may all vary together, whilft m and n remain inva¬ 
riable. And by exprefling the nature of the curve in fuch 
manner, many ufeful conclufions relating thereto may be very 
eafily inferred. 


4. Suppofe 


Chap. X. 


ANALYSIS. 


123 


4 - 

Suppofe the equation of the curve EFG to be 
ax*’y q bxy &c. = o : 

and fuppofe that, when v + mz and w + nz are therein re- 
ipedively fubftituted for x and y, the refulting expreflion, viz. 

to4-» 2 | ? 4- b.v-\- mzf . 'w + nzt &c. is 

= A 4- Bz 4- Cz' 4- D 2 ’.p g? + i. 

P+q denoting the order of the line EFG, and A, B, C, &c. 
being fundlions of v and w, in which z is not concerned. 
Then the value of 

mzf . <w "+ n ^ 7 ] + 

I/ — ^ v + nz\*~\ &c. when only z is confidered 

as variable, and unity is wrote inftead of [t _i_ z], being the 
lame as when v and w are confidered as variable, and z as^inva- 
nable, and m and n are wrote inftead of ft v] and ft w] 
refpettively; A, B, C, &c. will be found as follows. 

1 ft. From the above ■afiilmed equation, we, by taking z equal 
to Oj get ad*'ud -|- bv w 6tc. = A. 

2dly. By refidual divifion, making t — t the divifor, and, on 

a ^* urne ft equation, conftdering v and w as variable 
whilft z is invariable, and writing m and n inftead of ft _l v] 
and [/ w] refpedtively; and, on the other ftde, conftdering z 
only as variable whilft A, B, C, &c. are invariable, and writing: 
unity for [t^z]-, we get 

am P . v -J- mz 1 . w -j- nzf anq . v mzf. w -}- 1 

+ bmr . v+mz \ r ~\ w+nz] s -f bns . ^+^ r . w+nzf^kc. 
= ® + 2G2; + sD** + 4E2 1 &c. 

Hence, by taking z equal to o, we have 

amp^ w q + anqdbmrv 'w -j- bnsvw *~ x &c. = B. 

R 2 And 




THE RESIDUAL 

And the values of C, D, E, &c. may be found in the fame 
manner. 

It is obvious therefore, that, if, in the equation of the curve 
EFG, (the terms being all brought to one fide,) and the feveral 
afymptotic equations deduced therefrom by refidual divifion, (as 
in the preceding chapter,) v and w be fubftituted inftead of x 
and y refpedtively ; the feveral exprefiions, after fuch fubftitution, 
will be the fame as thofe by which ( z °, z, z\ z\ &c.) the 
feveral powers of z are refpedtively multiplied when v -f- mz and 
w -f- nz are fubftituted for x and y in the equation of the curve 
EFG, as before mentioned. And, by means of fuch exprefiions, 
we may not only determine the afymptotes of curves; but 
likewife, with the greatefi: facility, may find their diameters and 
centers ; and moreover, difeover many other remarkable particu¬ 
lars relative to the interfe&ions of lines. 

Example. Let the curve bepropofed whofe equation is given in 
Examp. 2. of the laji chapter : which equation is 

xy x + ey — ax' — bx z — cx — d = o. 

Then, tranfmuting that equation, as above mentioned, we 
have 

to* -{- ew — av* — bv* — cv — d 
4- mw" -[- 2nvw + en — 3 amv" — zbmv — cm x z 
-f- 2 mnw + nv — lajnv — bn? x z x | 

+ mn — am' x z' J 

Now, confidering m and n as invariable whilfi: v, w , and 2? vary; 
if z has but two values, and one of them is always as much 
negative as the other is pofitive, the line efg, with refj^dt to 
EFG, will be a diameter of the firft kind: Which will be the 
cafe when the coefficients of z' and z vanifh, and, at the fame 
time, the coefficient of z l and the terms in which z is not con¬ 
cerned, do not vanifh. Therefore, ^ being as determined by 

the equation mn — am' = o, let the equation of the line ef g 
be the coefficient of z put = o, i. e. 

mw 1 + 2 nvw + tn —< 3 amv — 2 bmv —• cm = 0; 

and 



I2 5 


Chap. X. ANALYSIS. 

and that line will be a diameter of the firfl kind, with refpetft 

to the curve EFG. Moreover, ^ having three real values when 

a is pofitive, there will then be three fuch diameters; which will 
be exprefled by the fame equations, by which the hyperbolic 
afymptotes of the curve EFG are exprefled, in the fecond Ex¬ 
ample in the preceding Chapter ; and, ef g being fuch a diame¬ 
ter, Yf will be parallel to a rectilinear afymptote of the fame 
curve. 

The equations exprefflng thofe diameters are 2inv -f- e = o, 
t w r -\- 2 a T vw -f - a z e — 3 av z — 2 bv — c = o, and w z — 2 a T vw 

JL 

— a z e — 3 av' — 2 bv — c — o. The firfl of which, when 
e is = o, exprefles two right lines, one coinciding with the 
bale, and the other with the principal ordinate ; whereof, the 
former is a diameter of the firfl: kind, and the latter an afymptote 
of the propofed curve. By the fecond of thofe three equations, 

w is = — a T v + 2 a 1 J v x -f- ^-v -{- — ; and, by the third, 

w 19 = a z v + 2 a* ,J v* -f- y v + C . It appears there¬ 
fore, that if i- be = c — a T e , or — = c 4 - the fecond or 
4 a \a 1 

third of thofe three diametric equations will accordingly exprefs 
two right lines j one of which will be a diameter of the firfl kind, 
and the other an afymptote of the curve EFG: and that, if e 

be = o, and — = r, the fecond and third of the faid diametric 
4 a 

equations will each exprefs two right lines j whereof, one will 
be a diameter, and the other an afymptote, as juft now men¬ 
tioned ; fo that, in this laft cafe, the propofed curve will have 
three rectilinear diameters of the firfl kind. 

If a be = o, the fecond and third diametric equations both 
become w l — 2 bv — c == o, an equation to a conical parabola j 
which differs a little from the parabola that, in the fame cafe, 
is an afymptote to the curve : and this diametric parabola will 
bifeft lines drawn parallel to the bafe, and terminated by two 
branches of the propofed curve. 


If 




126 THERESIDUAL 

If both a and b be equal to o, the coefficient of z will 
vaniffi when — is == o; which ffiews that no right line parallel 

to the bafe can cut the curve EFG in more than one point. It 
is therefore obvious, that, in this cafe, (as well as when a is 
negative,) the curve will have but one diameter of the firft kind, 
which will be expreffed by the firft of the three diametric equa¬ 
tions above written. 

It is manifeft, that, if the equation of the line efg be the 
coefficient of z* put = o, i. e. 2mnw n v — 3 am'v — bm~ 

— o; let — be what it will, (provided it be not a root of the 

n 

equation — — = o,) that line will be a diameter of the 

fecond kind, with refpedt to the curve EFG. 

And, — being of any value whatever, when b is = 0, w in 

the laft mentioned diametric equation will always be equal to o 
when v is equal to o j and confequently the point where v (or x) 
begins will, in fuch cafe, be a diametric pole. 


In finding a center of either kind, by our method, efg mull 
not be considered as a line, but as a fixed point; and therefore 
v and w muft be confidered as invariable, in the coefficients of 

the feveral powers of z, whilft and z vary. Now, as muft 

be variable, it is plain, that mn — am' , the coefficient of z , 
cannot be equal to o. Therefore, for the propofed curve to 
have a center of the firft kind, the coefficient of z muft vaniffi 

without determining the value of — ; and the terms wherein z 

71 

is not concerned muft alfo vaniffi, that the remaining terms 
in the equation of the curve may be divifible by z. Con¬ 
fequently, when a line of the third order has a center of 
the firft kind, fuch center will always be fome point in that 
line. Moreover, for (2 maw -f- nv 1 — 2 > am v >— bm ) the 
coefficient of z l to vaniffi without determining the value 


127 


Chap. X. ANALYSIS. 

of ^, it is evident b , v 9 and w muft each be = o; and fince, 

the terms in which z is not concerned being fuppofed = o, 
w cannot vaniffi when v vanishes, unlefs the term d be 
wanting; it follows, that the curve will have no center of 
the firft kind, unlefs b and d be each = o; and that, if 
the terms bx" and d be wanting in the propofed equation, 
the point where the abfcifla (*) begins will be a center of 
that kind. 

It is obvious, that, if the coefficient of z z vaniffies without de¬ 
termining the value of and, at the fame time, the terms in 

which z is not concerned do not vaniffi ; the curve will have a 
center of the fecond kind. Which will be the cafe, when b , v 9 
and w are each = o; and the term d y at the fame time, is not 
wanting: the point where the abfciffa (x) begins being then fuch 
a center. 

This Chapter might be extended to a great length, in expati¬ 
ating on the ufefulnefs of our method of tranfmuting the equa¬ 
tion of a curve; but what is already faid may fuffice to enable 
the intelligent Reader to make a farther application of it, at 
his pleafure. 


In the inveffigation of proportions, as well phyfical as geo¬ 
metrical, the relation between two variable quantities is frequently 
the object of our enquiry; and fuch relation never appears more 
clearly, than when one of thofe quantities is confidered as the 
abfci{fa y and the other as the correfpondent ordinate of a curve, 
and the form of the curve is properly afcertained.—By conlider- 
ing the variable quantities in that light, miftakes are prevented, 
and the moft fatisfa&ory conclufions obtained, in various intereft- 
ing difquifitions, particularly in the refolution of problems i elat¬ 
ing to the maxima and minitna of quantities.—Now, in afcer- 
taining the form of a curve from the equation thereof, the 
Articles in the laft three Chapters will, I prefume, be found of 

confiderable 


128 the residual analysis. 

confiderable ufe; and therefore be acceptable to the ingenious 
lovers of Geometry : Who, finding the procefles eafy and ex- 
tenfive, will, I am perfuaded, efteem their farther application a 
pleafant exercife. 


End of the First Book. 









































































































