[ «< J 


IV. On Infinite Series. By Edward Waring, M. D. F. R. S. 
Lucafian Profejfor of Mathematics at Cambridge. 


||ead December i5, 17S5. 

1. T N the Paper, which the Royal Society did me the honour 
X to print, on Summation of Series, is given a method of 

finding the fum of a feries, whole general term ((where 
P 

- is a fraction reduced to its loweft terms) is a determinate al¬ 
gebraical funftion of the quantity (») the diftgnce from the firft 
term of the feries, which always terminates when the fum of 
the feries can be exprefled in finite terms. 

2. The terms of every infinite feries mull necelfarily be 
given by a fundion of z, or by quantities which can he re¬ 
duced to a fundion of z. 

3. Let Q = A x A / x A /2 x .. A'” X BxB'x B' z x ... B'* X C 
O' X C'* x . • C /r x kc. where A\ A'% A' 3 . . . A'% are fuc- 
ceffive values of A; that is, refult from A by writing in it for 
z refpedively, z H-1, z -j- 2, z.+ 3, . . %-\-n; and B', B' • % B' • 
B'”, refult from B, by writing in it for z refpedively z + 1, 
z + 2, z + 3,... z,+ m ; but B is not a fucceffive value of A j &c. 
Let the numerator P = E. E / . E /z .. E 1 *- 1 . F . F'.. F /x ... F 1 • *r r 

x L; E', E 12 . . E Ii—I ; F / , F* •* . . F li—I , &c. denoting fuc¬ 
ceffive values of the quantities E, F, &c. refpedively ; and L, 
admitting of no divifor of the formula K x Kf where K' is a fuc- 
Vol. LXXVI. M ceffive 


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82 Dr. Waring on 

ceffive value of K; let L = A x B x C x &c. x E 1 * x F 1 * x &c. 
x p' x q' x r' x &c» — A 1 * x B Ira x C Ir x &c. x E x F x &c. x p x 
q x r x &c. where q', r\ Sec. are irrational quantities and 
fucceffive values of p, q, r. Sec. The favors A B, C, &c. E 1 *, 
F 1 *, &c. being given, the factors ft, q ', r', &c. into which they 
are multiplied in the quantity L will eafily be deduced by de¬ 
ducting the preceding irrational factors contained in A, B, C, 
&c. E lA , F 1 *, &c. from the correfpondent irrational faCtors con¬ 
tained in L; and in the fame manner, from the faCtors A 1 ", 
B 1 ", Sec. E, F, &c. can be deduced the irrational faCtors of the 
preceding p , q, r, See. 

Afl'ume for the fum of the feries fought the quantity 

_ ExE'xE”.. E'^’xFxF'. . F 1 * -1 x &c. __ 

A . A' . A 12, . . A IK—1 xBxB'xE 1-1 . . B Ira — 1 xCxC' . . C 1 • r “‘ x &c. 
xpxqxrxsx Sec. («%*' + 0 z m '~ T + + Sec.') = V ; where 

m! is a whole number, and«, 0 , y, Sec. are co-efficients to be 
inveffigated ; write in Y for z its fucceffive value ss+ i, and 
let the refult be W ; reduce the difference W — V into a fraction 
in its loweft terms, and make the co-efficients of the correlpon- 
dent terms of the refulting fraction = (W -V) and of the given 
fraClion equal to each other, if poflible; and thence may be 
deduced the fum of the feries required. 

4. This feries will terminate if the fum fought can be exprefled 
by a finite determinate function of % ; if not, it will proceed 
in infinitum , and may be expreffed either by a feries afeending or 
defeending according to the dimenfions of z. 

5. If any faCtor, A or B, or C, &c. have no fucceffive one in 
the denominator; or if the greatefi dimenfions of z in the de¬ 
nominator be greater than its greatefi: dimenfions in the nume¬ 
rator by 1, then the fum of the feries is not a finite algebraical 
function of z. 

6. If 



Infinite Series, 83. 

6 . If ill the denominator are deficient fome intermediate 
fucceffive fa&ors, multiply both the numerator and denomina¬ 
tor by thofe deficient factors, and they are fupplied: for exam¬ 
ple, let A x A /// x A ///// x &c. be factors of the denominator, 
in which are deficient the factors A', A' v , A f/ ", See. multiply 
both numerator and denominator by the content A / x A." X 
A /W x&c. and they are redored. 

7. If in the denominator is contained the content 

A'x A /f xA //( . . A /b, ’’ = H, in which X is the lead: of the in¬ 
dices /, /', /", &c.: affume A x X A /X x A ! ,h .. . A ibX " x« = H, 
and in the fame manner reduce the faftors in a ; then affume 

for the denominator A x x A A x A /A .. . A' x &c.: for exam¬ 
ple, let the contents be A 3 x A /s x A' n x ///S X &c. = H, then is 
2 the lead index, and confequently the content reduced as be¬ 
fore taught will be A 2 . A /2 . A" 1 . A ///z x a = A 2 . A /2 . A " z . 
A ///2 x A . A' . A /r/ x A /2 . A " /2 ; but between the factors A' 
and A!" is deficient the factor A"; and between the factors 
A /2 and A!" 2 is deficient the factor A" 2 ; multiply the numera¬ 
tor and denominator of the given fra&ion by the deficient 
fa&ors A " x A " 2 ; affume for the denominator A 2 . A' 2 . A"\ 
x A . A'. A" x A' 2 . A" 2 x See. = A 3 . A' 5 . A //s x &c. Sec. 

8. If the greated index of the content H is contained in one 
factor only, then the fum of the feries cannot be expreffed in 
finite terms of the quantity %. 

9. The fame may be applied to the contents of the feveral fuc¬ 
ceffive values of the quantities B, C, &c. in the denominator: 

for example, let the general term be t ^ 2 , g ’ multi " 

ply it into z—j to compleat the deficient term, and it refults 

M 2 1 



84 Dr.Wk^mi. 

-— ~~ —alfume, by the preceding method for the fum of 
the feries the quantity u . - . . of which the fucceffive 

* J 1 .£ . 2 . . Z-~ l 


term is 


I • 2 • 3 

I 


", and their difference----- - — 

S X.2..Z—I 1.2.. 

the given term. 




1 1.2..Z —2 X z 

2. Let the term be Ne* and e lefs than i, which is the term 
of. a geometrical feries; then will the fum of the infinite feries 

he 2 i-xe% beginning from the term whofe diftance from the 

firftis z; for the difference between the two fucceffive fums as 

** Ne* the given term. 

2. Let the general term be —- 4 ; —- x e *+ z ; alfume 

for the fum of the feries the fubfequent quantity (z + i . z +z .. 
% 4- 3 . . xe x y. (a + jSs + ys 1 . . Z*- 1 ) and by the 

preceding method the co-efficients «, / 3 , y, &c. may be found : 

the fum is known to be = - 4 - X e z+I + X e* +t X e a + 3 


jsj 4* i 




* + 3 


;, JL-e x +” f which can eafily be reduced to the preceding 
»+» 

formula. 

If the general term be q+p or " ( qq^" 5 w ^ ere 1 * an ^ T / , 
P and P', Qjand.Q^, are fucceffive terms; then will the fums 
of the feriefes be — or— properly corrected. 

io. If the function expreffing the general term contain in the 
denominator a faflor or factors, which have no fucceffive one ; 
reduce the faftor or fa&ors into an infinite feries,proceeding 
according to the dimenfions of z, and thence, by the method 
before given, find the fum of the feries. The fame method 
4 ma y 



Infinite Series. 8 c 

may be purfued, when the denominator of a fluxion, which is 
a function of x multiplied into x contains the Ample power 
only of a fador or fadors; reduce the fador or fadors into an 
infinite feries, proceeding according to the dimenfions of x y and 
by the known methods find the fluent of the fluxion. 

11. The fluent of the fluxion or fura of the feries may be de* 
duced alfo from the fubfequent propofitions, from which may 
be inveftigated many feriefes, whofe fums are known. 

1. Let/P = Qy, qQ = Rr, rR = Sr, jS = T/, &c.; then ^Pp a* 

Pp — + Rr — S s +Tr — &c. if only the feries converges. 

2. Let pP'+p'P'-Q^q', q<X+y'<%+=Rr', rR' + r'R'- 
S/, rS / + /S'=T/, &c. where P', p', Q. / , q', &c. denote the 
increments of the quantities P,/>, Q, q, &c. reipedively, then 
will the integral of the increment (Pp') = Pp ~ Qq -f Rr - Sr + 
&c. if only the feries converges. 




-i- ( —h- + A + 

\m +1 r' + i s '-f I ^ t'+j 


C+^D + &c.) ===1—; whence -L- ~ _L_ + x 

T+x ' i ~nx n ~ x I-» »a+i r'+i 


I 

m + I 


»+ r’ n + nt i 

. ^ mmm m . X 


i '+1 r'+i m+i 

q—x r 'x, Q = x~"~ r \ r — x s ‘x. See. 
Ex. *. 


f See. In this example p==x m x, P~x~”~” 


* ^ f x m ~~ n ~~ r X 2Z j ^w—»— 

J m — n — r+i 


r*f 2 


+ x 

V»-+I iW+I 


«+r 


«*fr a+r+i 


+ &c.); whence 

m+ 2 , m 4*1 m + 2 , m + % ' 

— . 1 _= —+ +”±I±-?B + &c. In both thefe exam* 

n—r + 1 m +1 2 m 4 * 3 

pies the letters A, B, C, &c, denote the preceding terms. 


Ex 3 r*-t =“—( ■ 1 
'3'J i+* i+a«+^ 


X 

■X —-+ 


m 


+ I.OT + 2 * + # 02 +I .ZB + 2 . OT +3 


X 


*" ,_ l • 3 _ # -f&c.) 

(i +*)*" 1 "w + i • m + 2 • »*+3 • ffl+4 * C 1 +*) J ^ 


Ex. 



86 


<( 


a'x m x 


Dr. Waring on 

X 


«V+* 


:.«V + 1 


a + to“ + »»+'l (a + bx n +cx* n ) h + «■+ i . (a + 

+ &c. 

> ;/2 + » 4 I /» 4 2« 4 I / 

12. Let the general term of an infinite feries be 

_ A _ ' 

% + » . 2 + a+l . « + <H- 2 * • » + « + »X2-1-|3 . JS4#4 I « %404$ • • *+£ + 

jwx«+y . x + y+i . %4y4 2 . . 2+y+rX2+17z+^-(-1 . as + ^+2 ,". * 
—r- r—; where « — j 3 , a — y, fi-y, &c. are not whole 

K + HjX&C. 17 11 ' ' 7 

numbers; the fum of the feries can always be expreffed 
in finite terms; if the fum of the fractions 


i , 2 .3 .. n 


0 — oo . 0— c*4” I . 0 — a 4- 2 ♦ . <3 — a 4- m X?-a . y — a-J- i . y —<* 42 V* . + r 

I 

—----— -—---.-— — . .. — - ——— X 

X ^ — oo • $ — 4 I * $— a Hh 2 « • $—* a 4^ X &C» XXI • 2 • 3 v • a—* I 

_ I 

0 — a — I • /3~ oo % 0— a+1 . /3 —a4" 2 •. ,0 — a + zrc— I Xy-«- I . y — a.y — «4~I 


. . y — a 4 r —• I X ^ — « — I . $ —U . £ — a -f- X , . i'— a -f- j — I X &C. 

I I 


r + 


I .. 2 X I • 2 • 3 • . — 2 0 - 00—2 . #—*-I . £-* . g-»+I • • £?-a4»-2 Xy-« 

— 2 . y -a- I ,y-a . . . y~a4 « — 2 X a— 2 • £-a-I . a . . + x &C # 

- ^_ 1 _x — — _ 1 . ... 

I #2 » 3 X 1 • 2 • • W — 3 0—00 — 3 , £ —«— 2 • .0— a+ »-* 3 X y — a— 3 * y— 

4 " »*» —t — r— ““ X 


e&—2 . • y —a + »— 3 . ^-«~2 . . ^-a + J-3X&C. 1- . 2 • 3 . . 8 

■ _ I _ ~ 

g-a-« , 0 ~.co— 8 4 I . . 0— «-~» 4 ,/n Xy—*—» . y — a — « 4 * I ».y-a-« + rX 

I w I 


§ — & — n • 00 — a+i'n^*—a^JXSce. 1 • 2.3 . . /w ,N <*—0 . a — $4* 

I • a — ^42 « • co — 0 ?fl X y — 0 • y — /3 4 I • « y — 0 4 “ ^ X 0 • $ — I 

I I 


X 


-/S + 2 . . 0 + JFX-&C. I X I . 2 • 3 . • J»— I 


-I .«-P. .«—!?+ 
■«.— I 



8 7 


Infinite Series. 

i __ 

u ***** I xy—0 — l . y— @1* .y—0-f- r —i X^—0 — i -0 ,. £—0-J-j— I X§&c. 

_ 1 _v/_____ 1 _ 


I 2X1 .2. 3 . . /72 —2 a —0—2 . a —0—1 . a — 0 .. « — 0+ «— 2 X y — 0— 2 



w+i . • y—0—7K + r x ^ —0 —/ 


•0 — ZW-f' I . • ■$ — 0 — 7/2 + ^X &C, 


I u. HI .. w . ■ nr III ITU ^ — " ------— -1 . it ~*~l w ~ it-w ~ i rmnrirmr i ifri W i 

I. 2.3. r a “~y • y + 1 • • y+« X 0 —y . 0 —y-b I . 0 — y 4- m 

- X )-,.»->+, .T7^h^ ~ & c - = °’ &c '=° ; or > to explain 
it otherwife, affume the quantities a, a+ i, oj + 2 , a+ 3 , . . . 
«34 + tf ; fi» /3+ j, /3 + 2,. . /3 + 772; 7, 7+1, 7 + 2, • . 7 + r; 

J, J“+1, 5+2, .. S + J; &c. fubtraft one a from all the re¬ 
maining quantities 05 +1, a+ 2, . a + n ; / 3 , /3 + 1, . . j 6 + 7 / 7 ; 
7, 7+ 1 * ♦ • 7 + 7*; d, £+ i, .. £+j; &c. and multiply all the 
differences refulting 1,2, 3,.. »; /3 — os, /3 — a + 1,.. fi — tx+m; 
7 — <34, 7~a+I,..7 — a+2, *.7 — «+r; ^ — a, §— 05 + I, . . 

S —a + j, &c. into each other, and call the content^. In the 
fame manner fubtraft 05 + 1 from all the remaining quantities «, 
05 + 2 , 04 + 3 > ,,c 5 "h 77 j $» / 3 +l,.tj 6 + 772 j 7, 7+I,..7 + rj 

§ + 1 ,.. J+j; &c.; and let the remainders - 1 , 1 , 2 , 3 , • • » — i» 

/3 06 ” I, /3 — 05 , jS“(35+1, • • $ — 05 + 777 — I ; 7 — 05 — 1 , 7 — 04, 

7 —0!+I,..7 — 04 +r— l Ci — I • $ — 05 ,$— 04 + I,».$— 05 + S — I, 

&c. be multiplied together, and their content be called /“+V 
In the fame manner fubtraft 05 + 2 , 05 + 3 , . . 05 + 72 refpe&ively 
from all the remaining quantities, and let the differences re- 

fultins 










88 Dr. Waring on 

fulting be multiplied together, and their refpe&ive contents be 
called . « p *+ n » then, if the ham of the feries 

can be found, will —-f—4- + ——■ 4--4- ... —^-=o. 

pQC, />* + I 2 ^4-3 pK~\~n 

In the fame manner fubtraffc / 3 , jG + i, /3 4- 2, .. @ 4- m re- 
fpeftively from all the remaining quantities, and multiply 
their refpeftive remainders into each other, and call their con¬ 
tents refpe&ively . . ^+ w , then will 4. _L_ ^ 

— « • "^77" ' Tir o# 

p0+2 pP + m 

Subtract y, y+i, y+2, .. y + r refpe&ively from all the 
remaining quantities, and multiply their refpedlive remainders 
into each other, and call their contents refpeflively py, pv+* f 

i>v + % . . pv +r ; then will —-j—-—1—— . . —= o. 

r 1 py py+1 pyH-2 pv+r 

Subtract $ H-1, $+ 2, . .J'+j from all the remaining quan¬ 
tities, and multiply their refpeftive remainders into each other, 
and call their contents refpeftively p* t ; then 

will -4 4 ———|—4—f —— = o; and fo on ; and, vice verfd , 

if the fum of the above-mentioned fractions be refpe&ively = ©; 
then the fum of the feries, whofe general term is the given 
one, can be found; other wife not. 

13. If the fum of the feries, whofe general term is 

1 


• 56 -J“ a 4 ’i • • z-\~ n X • z ~t j 3 -i- 1 • • fft X Z'jry • 1 • * 

————:• .», ——ttt— r;--5, can be found in finite 

terms ; then the fum of a feries, whofe general term is 

(where / denotes an affirmative number) 


can be alfo expreffed in finite terms. 


14. Let 



1+LetA X55 + /S . 55 4 -y • 3 +"?X z+eX &c. +Bx % + a X » + y . 
5 + 1 . Z+£ . &C. +GX 55 + C8 . j s + jGx^ + L. » + s • &c. + Dx 

»'+«. » + /3.5S + yXZ+6 . &C. + E X5S + 05 . Z + /3 .Z+y.Z + l 
X&C. = 1, whatever may be the value of then will 


y-« *♦ &c.’ • y~$• .&c.’ *->"• &-y .■—y• &c* 

D=— g—y r * -r-r—» &c. 

15. Let the general term of a feries beg- x e*, where e is lefs 
than i ; the fum of the feries can be exprefled in finite 

1 1 

terms, when the above-mentioned quantities ~ “ 

* . * p Ot pot -J-1 pot -J- % 


g* g -I 0 C 0* ■ c 

p<*+ 3*' ”^*/>«+» ^0+1 2 "^*^0+3 p@+m ° J 


1 1 


1, ■ * 
pY^ py+l^pY+Z 


e r __ 1 e ? e J 

^y+r * /• ^ pb-\-1 /}+2 *^”^+ 3 


~i— ss o, &c. = o. This never happens unlefs * * 1. 

jf>}+S ’ 

16. If the general term be any rational fundion of z into the 

• | • Gzi^bz ^ cz> 

exponential **, wz. . 1 — • - T ~ " 

J5+« XK + e + 4 X Z + Ci + k X &C, XZ + 0 * 

. ■- -■■ - . . .I.. . ... ... .. ■ . . . . x f* = K 

5+0 + h' n X 8+0+V"' X&c.Xss+y - X z+y+V' r Xz+y+i" r X&c. ? 

where A', A 7 ', &c. i, k'\ See. I, m, m\ m". See. n, 

&c. r, r', r"\ Sec. See. denote whole numbers, and neither 
Vol, LXXVI. N # - B 




9 ° 


Dr. Waring on 


a- (2, nor a-y, nor jG - y, &c. are whole numbers * let m be 
the greateft of the indices m, m\ m", &c.; n the greateft of 
the indices n , n\ n'\ &c.; r the greateft of the indices r, 
lorn the fums of the feriefes given, whefe 


&c.; then 
terms are 


general* 

O 




(Z "f” to ) 2 % Ci 
e* 

9 &c. cam 


^+(3)’’’ (z+/3)” 1 !5+ ^ (2+y) r (z+y)’ 1 *+y 

be deduced the fum of the above feries, whofe general term Is 
given above ; multiply each of thefe terms into unknown co¬ 
efficients e',f, g, b, &c.; then reduce them to a common deno¬ 
minator, which is the fame as the denominator of the given 
general term, and add them together, and make the correfpon- 
dent terms. of the fum refulting equal to the correfpondent 
terms of tlie numerator az ! + fa 1 - 1 + cz l ~ z -f &c. of the given 
general term; and from the equations refulting can be deduced* 
the co-efficients e /, g, h, &c. and. thence from the given fums 
the fum of the feries required. , 

Approximations to the fums of the feriefes may be deduced 
from the methods given in the Meditationes Analytica. The fum 
of fome few cafes have been given from the periphery of the 
circle: for example, when a and m are whole numbers, and 
ezzj ; or, more particularly, when m-z and <?= £,' and 
fame Other particular cafes, which may be with nearly the 
fame facility calculated* from approximations; the cafes given 
indeed are. foTew,. wilefts when- e=? x, that they.can very rarely 
be applied, 

i jr.;If the dimenfions of z in the numerator be equal or greater 
than its dimenfions in the denominator; that is, l be equal or.- 
greater, than m+v" 



Infinite Series, y i 

+ &c. &c* reduce the fra&ions to a mixed number, fo that the 
dimenfions of z in the numerator of the fra&ion be lefs than 
Its dimenfions in the denominator, and the integral part be 
Ae* + Bze* + CzV* + DzV + ... Hz'V: the fum of the infinite 
feries whole term is Ae* + Bzr* + Cz V + Dz V + Ez V -j- . . . 


.+«*■-£ ,+B - ji.) + C<^,_ + D, 


of a feries (whofe general term is z m e*') = LsJ lz Jl iAs SSL e 

IqW~I _ TgW—I . TgW —2 __2 M 1 __ 

-—I . a . 3 . . m - xe + -? ■■ X i. a ,3. m-z . g 


(i-*) 


— X I ♦ 2 • 2 • , tn — h — i» . t±z 

v—h 


{l-e) 


;, where Lis 


equal to the fum of all quantities of the following fort, 
•S’* -1 X / 3 S"‘~ a ~ I x v S m ~ a ~ li ~ I x * x >$«—«■—i&-r-s~i x 


where a, /3, y, &c. are whole affirmative numbers; (in the 
preceding notation by ’ r S ? is defigned the fum of the contents 
of every tt of the following numbers i, 2, 3, 4, 5, . . . ^ • 
and a + (3 + y ■+ s + &c. = h + 1 ; the above-mentioned pro- 
du6t “S ’* -1 x 1 x &c. is to be taken affirmative or nega¬ 

tive, according as the number of letters «, (3, y, J, &c. is even 
or uneven. 

The fum of the feries z“ x e % may alfo be found by afiumihg 
for it (az”‘ + bz m -' + cz’ n -* + dz m -3 . . . k)e* ; then, finding its 

fucceffive term (^xz+i f + fe+i e+cz+i e + &c.) 
and taking the difference between it and the alfumed quan¬ 
tity, there refults (axe — 1 z m + mac -f be - 1 z m ~ l + &c.) e "; 
by equating it to the given term-z’V* are deduced the fub- 
fequent equations axe - 1 = 1, mae-\-be~ 1 =0, &c. whence 

N 2 





jDr.WARiNG m 


f *' 

a = —£ = ~ - &c. 

«— i e 

If e— i, then affumc <?ss“+ r 4f fe“+&c. for the fum fhught^ 
which rule was firft taught by M. J. Bernoulli. 

If the term be z”’f b *iotf b fubftitute e r and there refults, 
% rn e a the fame as before. 

18. Let P * A + Bx n + C at 18 -p E& 3 * q*. &c. then will 


the fum of the leries ~—g~ ^ y i ■“ d- ■ ”.r ; B * ■; .r~ +, 

a • p • y • 8 .\■ «■+« . P+# ty-J~» • occ. j 
C**".— 4 .&C -. 1 . y p_i _i_ JL : 

*+2» ./3+2« . y+,2» .,&C. ‘ * « * |2—« *. y—ot, * 


— . Xi&,Q. X- a J-X u £- 


&.C. A—' 5 


— J■"* (S’.-u Vf ^ ^ 

i_ . _i- . —— . _1_ . &c. *xrv-Jfjwp-&a 

This may be proved, from the fubfequent arithmetical propo¬ 
rtion— • t—* • &c.+ 4 ~ ,■ , -i-. &c.-p- 


« — y # —y ^~*y 


. -d— . &c. 4 ?- 1 . 


Ofc-—y-<-£ 


• &c. =s 


__ i- 

« . (3 . y . £ . &c. * 

19 ..Let- the general term of the above-mentioned feries A + 
’«f-Cy®+&c. be Hi*”’then from the.Turns of the feries and 
the fluents of the fluxions x a p,, x^j>, xvj>, x*p :> &c. being given 
there follows the fum, of a feries, whofe general term is 

>-7— ~ Vr~~— x ~~TZ~~~T T‘ x fL®",.. where. / denotes a whole 
a+nz . @+n% .,y+»«,. ^+.» z; .-.See,. »- ' 

number... 


If w . 


m~-*i m-%. 


m—*~i m — 2 




, . where / is a whole number^ and tn, af, 0 , y', &c. are 


either, whole numbers or fractions whofe denominator is 2, and 
a ^ an, J 3 — / 3 'n, &c. the fum of the above-mentioned feries can be 
found by finite terms, circular arcs and logarithms* II 



Infinite Series. gj 

^ H== i. 2 * 3 . . * or “ 1-2 *3 -. /«» atld k** &e. whol* 

numbers and 0 == 1 ; then can the fum of any feries of the above- 
mentioned formula be found in finite algebraical fun&ions of 
and the circular arcs and logarithms of them. 


imi*. 


F A R T III 

k THE doftrine of proportional parts was probably very’ 
early known in the asra of fcience; for when men could not 
find the exa& value of a quantity, they were induced to find! 
near approximations by trials, and from thence, by proportion, 
an approximation Hill nearer: which method is commonly 
denominated the Rule of Falfe. 

This was often found to deviate confiderably from the exad 
value; and the fame operation was repeated, which frequently 
produced a nearer approximate value, and fo on. 

This method of approximations, the moft general yet 
known, has been ufed in refolving problems by feveral of the 
moft eminent mathematicians in different ages, and in this 
particularly by M. Euler. 

2. The following obfervation, I believe, was fifft publifhedi 
in the Meditationes , in the year 1770, viz. that the conver- 
gencyof the approximate values, found by the rule of falfe 
and method of infinite feries,, generally depended on this, viz., 

horn 



94 Dr. Waring m 

hovf much nearer the approximate ahumed is to one value 
of the quantity fought poffible or impoffible than to any other, 
and not to the quantity itfelf: hence, when two or more («) 
values of the quantity fought are nearly equal, it is neceflary 
to recur to more difficult rules, "‘ofa. to three or more trials; 
as, for example, let two roots be nearly equal, and write 
a, & q 7T, and a + p, for the unknown quantity in the given 
equation made — o, and let the quantities refulting be A, B, 
and C, then will more near approximations to the two roots 
nearly equal of the given equation be a + the two roots(*) of the 

quadratic /— + — B -'■■■ C / , A~r+f 

A \ VT p vr(‘ 


•+ 


d*’- 


X 




•p.) 18 p(p — w)J 5 ” p) 

_—— ) x + A~oz for write o, tt, andp, refpeftively for x in 

the equation, and there will refult the quantities A, B, and C. 

More generally, fubfoitute for x in the given equation the 
quantities a , a + it, a+p,a+<r,a + T,a + v, &c. where tt, p, cr, r, 
&c. are very fmall quantities; and let the quantities refulting 
be A, B, C, D, E, F, &c, ; then will more near approximations 
to the («) roots of the given equation be <7 +a, a+ 13 , a + y, 
a + l, &c. where a, / 3 , y, See. fee. are the n roots (y) of the 


(*— «•}'(*— (V—cr) {b — t V&r. 
:±:?r£ or t* See* 


x Aq 


— t)&C. 


given equation — .. 

^B+-y=#^^^C + &c.=A-fl-?V + fA + 

—91*) Cf — \ 7T 9P / \ 


B 


X <? 


w(?r—— tt) / 

__—- r ) x e 


— ■7r)( f — cr) (g — -r) See. 

■•■e-r-w • e — ^ —<r 


£ — 7T ■ 


(; 


A 

e 0 * 


B 


e — & 


9T 7T - 

A 


—- ( a 

f -p+ (— 

D 


q 


f (f—IfJCf-O- 

B 


^(sr — §) («* — 0*) (&* *- T-) 

E 


* ( ff — T >{* — §i{* — t)&C. 
/ A B 

\ TTgOTW 


t(t ^(t —§((r— cr)&C 

c 


:) 


n(v—z)(v — <r)(ir~T)(v—v) £(?-«•) ((> — *) 

(c~ T ) 



Infinite Series * 


9$ 


D 


(?—O (e- 


■«) 
F 


or(cr— w) (<t — §) (cp— • t.) (fir — v) 

S\e. e -Tt . e-p 


-t)(r — cr)(T — v) 


e — tr . e — r + &c. =r o. 


^(F —* ( u (° Cr)[v — t) s 

Their refolutions were fir ft given in the M'editationes Ana ** 
lytica ,. published in the year 1774, and require the extra&ion 
of a quadratic, cubic, and' in general of an equation of («) 
dimenfions ; which rules will often, give a nearer approximate 
than the. preceding, when the roots are not nearly equal. 

3. Thefe rules may be applied to find approximations to the 
roots of algebraical equations : for example, let the algebraical 
equation be x n —px n ~'-\- qx n ~ 1 ~ &cc. = q, fubftitute in it for x 
two quantities a, and a + e much nearer to one root than to any 
other, and there refult a” — pa”- 1 + qa n ~* ~ &c -1. A, and (a + ey 
, —p (y + e)”~ 1 + <? (** -fc e ) n ~ 2 - &c.,r= B; then,, by the rule, of 


falfe, B — A : e \\ K : 


, »—-I , n—z o 

-pv -hqa — 


b\ 


(na 1 — n —■ ipa 2 + n — %qa n 3 — ) -f- &c a 

whence a — b a near approximate value to the root fought. If the 
quantities,in which are involved e,e z , e s ,ikc. on account of I? being 
very fmall, be rejected* then will the approximate fought ^ = 

; which will nearly be the fame 


n , «—i , #— Qrn 

a -f qa —&c. 


n —I . > 

na ~~n—ipa 


4 - n 2 qa . 

as found, where a near approximate is given, from the method 
given by Vieta,. Harriot, Oughtred, Newton, De 
Eagny, Halley, &c, 

4. From this exprefllon it follows, that if (ii) be a root of 

the equation na”~ l — n - i pa n ~ 2 + &c, o, of which the roots 
are limits between the roots: of the given, equation, the. ap¬ 
proximation found will be infinite. 

5. In finding thefe approximations,, when there are two or 
pore quantities contained in the given equation dependent on 

each 



g 6 Dr. Waring on 

each other, as the arc and the fine, it is neceflary that both 
fhould be corrected in every approximation to fuch a degree as 
the fubfequent approximations require. 

6 . In the Mediiationes it is obferved, that in any algebraical 
equation #" - /WT"* +, — cx”~* + dx n ~* —. ex n ~s .. d±gx ”- m + 1 =& ' 

hx n ~ m r±ibx n ~^ t z£k n ~*~ 2 ^&c.=:o, if a be much greater than 

—, and - has to r a much greater;ratio than a.i - j and in the 

a a b ■ • . ( : a 

fame manner | has to ~ a much greater ratio than b ~ : f, and 
fo on;; then will a .be a nearapproximate to the greateft root 
of the algebraical equation; - a near approximate to the fecond; 

1 a neari approstima:te to-the thirds and ^ a near approximate to 

a root, which is much lefs than m roots of the given equation* 
but much greater than the remainingj(» — m— i) roots. 

If the equation above-mentioned tf*— ax "- 1 q- &c. =r o, or 

a b 

which is the fame, i =*= ~ +-r=t:&c. = o be infinite; then will, 

X X 

in like manner, all its roots- be poffible- apd their approximate 

values a , - , c - , &c. as before. 
a b 

This eafily appears by fubftituting for a> b f c, &c» their 
values in terms of the root of the equation. 

y. Anearer approximate to the above-mentioned root will be 



8. Equations, of which the fluxions of the quantities con¬ 
tained in the given equations can be found, may be reduced to 
infinite algebraical equations, in which is involved no irrational 
function of the unknown quantities contained in the given equa¬ 
tions by the incremental theorem ; viz. let Asi o be the given, 
5 equation, 



Infinite Series. 97 

equation, and (a) an approximate much more near to the root 

required (7r) of the given equation than to any other: write a for 

(x) the unknown quantity fought in the fuhfequent quantities, A, 
• •••*« 

-*> —, &c.; and let there refult the correfpondent quan¬ 
tities oc, y, £, &c.; then will v — a be a root ( e ) of the infi¬ 
nite equation a±:Qe + —— ye z ~— £<? 3 + -— e See. 

* 1.2 r 1.2.3 1.2. 3 • 4 

= 0, of which a root of the equations ct,—\ Sfrro, adt ( 3 e + 
~^—ye 1 ~o. See. will be an approximate. If two roots of the 

given equation are nearly = a , then it is necefTary to recur to 
an equation not inferior to a quadratic. 

9. The fuccefiive approximate values found by thefe and like 
rules will ultimately be to each other in a greater than any geo¬ 
metrical ratio: for example, let be an approximate to a root 

of the quadratic x 1 - (hr+ + = 0, then will the new ad¬ 

dition to the approximate to the root s found by the common 
method at the difhmce n- 1 from the firft approximate be 

nearly, where b= 2 n ~ l . 

a 

1 o. Let an equation x n — P *”— 1 -f Q^— 2 — R #*” 3 4- S-t” -4 — Sec. 
*= o, of which the roots are a , b , c, d, See.; and an equation 
x n —px n ~ l + qx — rx ”~3 + j#” -4 — &c. = o, where p, q , r , s. 
See. differ from the co-efficients P, Q, R, S, &c. by very finall 
quantities : aflume the (») equations 7 r+p+ <r 4 -r+&c.=^ — P 
= «, aTr + bg + ccr-i dr + Scc.~Pu — q +Q~fi, a 2 vr + b z p + cV + 
d z T + &c. = P /3 — Qa + r - R = y, a'ir + b 3 p + <rV + d 3 r + &c. — 
Py — Q .|3 + Ra — r + S = J 1 , &c.; and from them find the un¬ 
known quantities tt, g>, <r, r, &c; then will <? + 7r, £ + c + <r, 
4/4- r, &c. be nearly the n roots of the equation x n -px n ~‘ l + qx’'-* 
Vol. LXXVI. O 



Dr. Waring on 


-&c.=c,and confequently n.s,&e.nearly- 

»«" 1 —n—ipa" 2 -fSec, 

b*-.pb n - I + gb”- 2 -&c. a , . r ' 

——■==~-^ =r -, &c.; whence the eonvergency of the ap- 

nb n 1 ~~n — ipb n -f&c, * 1 

proximate values found by this rule depends oil the principle 
before delivered* 

10. Let there be given (jri) equations, which contain m 
unknown quantities x f y, z. See .; and let a, / 3 , y, Sec. be 
nearly correfpondent values of the unknown quantities x, y r 
Sec. refpectively : aflume n i different values of the quantity 
x, viz. ct, a + n, a+ 7/, a + n", &c. Sec. ; and in like manner 
a flu me n -}- i different correfpondent values of the quantity y,.. 
which let be [ 3 , (3 + £, (3 -fp / , 13 4 - p", Sec. ; and . 1 © of the remain-- 
ing; where tt, n' r n", &c. p, ph p 7/ , &c. &c. are very fmall quan¬ 
tities; fubftitute thefe quantities for their refpe&ive values in the 
given equations, and let the refulting quantities be A, B, C, 
D, Sec. in the firft equation P,. Q, R, S, &c. in the fecond, 
&c. &c.: aflume from the firft equation the n fimple equations 
an + bg + Sec. =B~A, an + b/ +Sec. = C— A, an" + b^' *f&c, 
= D —A, &c.; and from the fecond equation the n fimple 
ones hn + &(> + Sec. — Q — P, hn r 4 - kj/ -f &c. = R — P, hn" 

+ &c. — S — P, &c. From thefe equations can be inveftigated 
the co-efficients a, b, &ec..b, k, &c. &c.,; ultimately aflume the 
m equations A-\-ae-{-bi + &c. = o, P y-he ^ki-^ &c. — o, See.. 
from which can be deduced the values of the quantities e, 
he .,; and « + r, /S +/, See. will be more near values of the 
quantities, x, y, See. 

n. Sir Isaac Newton found the fum (A) of the zn th 
power of each, of the roots of a given equation, and then ex¬ 
tracted the 2 « th root of A, viz. X/A for an approximate value 
of the greateft root of the equation, and further, added fome 
fimiiar rules on the fame principle,. la. 



Infinite Series. 99 

In the Mifceil. Analyt. and Meditationes the fame principle is 
applied in different rules for finding approximates to the 
greateft and other roots of the given equation; and alfo limits 
of the ratios of the approximate values of the roots- found by 
thefe rules to the roots themfelves are given. 

It is obferved in the Meditationes , that from thefe rules in ge¬ 
neral to find the greateft root, it is often neceflary that the greateft 
poffible root be greater than the fum of the quantities con¬ 
tained in the poffible and impoffible part of any impoffible root 
of the given equation: for example, \£ a ^-bs/ — 1 be an im¬ 
poffible root of the given equation, then it is neceflary that the 
greateft poffible root be greater than a 4- b. 

It may further be obferved, that in equations of high dimen- 
fions (unlefs purpofedly made) it is probable, the number of 
impoffible will greatly exceed the number of poffible roots; and 
confequently thefe rules moft commonly fail. 

12. M. Bernoulli affirmed a fraction whofe numerator is a 
rational fun&ion of the unknown quantity, and denominator 
the quantity, which conftitutes the equation ; and reduced the 
fraftion into a feries, whofe terms proceed according to the 
dimenfions of the unknown quantity ; and thence found an 
approximate value of the greateft or leaft root of the given 
equation or its reciprocal, by dividing the co-efficient of any 
term of the feries refulting by the co-efficient of the preceding 
or fubfequent term: for example, let the equation be 
a” - px n ~ l -f qx n ~ z - rx ”—3 4. sx n ~*.. = o; af- 


fume the fra&ion 


fix 


»??'— ipx 


m S&r | W®" 

4 




^-4 


4 -&c. 


-px” 1 + qx" z - 


•rx 




fix 4 " /3 + y 4 * $ 4 * &£•) x % 4* (jx 4- 13 2 + 4 * + & 4* 

4- 4* 4* 4 s $ 3 4 s s 3 4* (j% 4 4* /3 4 4* y 4 4~ $ 4 4~ s 4 4* 

O 2 &C.) 



joo Dr. Waring on 

6 cc.)x-s .... -f +/S”- 1 + y n ~ x + &c. = P)*-" + («’* + l 3 m + y 

+ &c. = Q.) ^- W “ I + (a w +' +• fi*+ l + y n +' + &c. = R) *-«-» + 

&c.; then will ^or be the greateft root nearly. 

•r-, R—2Q* + 3P^ a — 8fc. . . . nx n ~ l / I I i , » v . 

Ex. 2. -——----- = ( - -\ -h - + &C.) + 

S~R*-{-Q* -P* J +&c. . \ «■ $ y J 

(;* + |'.+^+T- + &c -) J: +( 5 - + p+^- + &c 0 ^+ (j+|i+ 

i-+&c.)*’ + ..... +(-±+-±- + - 1 -+ See. =P) *-= + 

/_L + JL + ± + &c.=Q)^- I + /'-^-+~V + - 4 r + dcc^^-fe.’ 

e m v m J / 3 m+i »*+■ J 

&c.; then will ^ or \j ^-, the leaft root nearly. 

V 'Sr 


Ex. 2 . ----=*-“ + (a + jS + y+ $ 4- &cc.yx~~ n ~' t 

x n — px n -f qx n z — &c. 

+ (,<#* +/3 2 + y 2 + &c.( + «jS + «y+$y+«^+&£•)) •v~ 3 ~ 2 + (a 3 -p 
0 3 + y 3 + &c .(+« 2 /3 + « 2 y + / 3 *y + y*a + y 2 /3 + &c.) + #/ 3 y + + 

«y§ + {3yS + &c.)) ^-«-3 -f- &c.; and — : -f-;= i f- 

' ' \ JJ S-R* + Q/~&c. . . x n 1 


(^ + j + 7 +&a > + ( 7 + F + ? +&c ' (+ rs + 4 + S +to -» 

x 2 + &c. in each of which all the numeral co-efficients are i. 
The approximate values to the greateft and leaft root may be 
found in the fame manner as before. 


From the preceding examples it appears, that the fame ob- 
fervations which have been applied to Sir Isaac Newton’s 
method are equally applicable to M. Bernoulli’s. 

13. In the Meditation.es this, rule is further extended, viz. 
let the given equation involve irrational and other functions of 
the unknown quantity; reduce it fo that no function of the 
unknown quantity (a) may be contained in the denominator, 
and let the refusing equation be A = o. Affume a fradion 




Infinite Series, mt 

~, whofe numerator B is a finite rational and integral fundion 
A. 

B 

of the unknown quantity; reduce — into a feries proceeding 

according to the dimenfions of the unknown quantity: for 
example, let the feries be AV + BV+' + CV+“4- . . . + 
hx r + h + + N^+ , '+ iJ + &c.; then ( exceptis excipiendis ) 

if s be negative, will the greateft root be ^ nearly; but, if r 

be affirmative, \j~ will be the leaft root nearly. If /be infi¬ 
nite, then (exceptis excipiendis , as before-mentioned) the quan¬ 
tities V! and \J~ will be the above-mentioned roots accu¬ 
rately. 

Thefe principles have been applied to find the remaining root? 
of the given equation as well as the greatefi; and leaft. 

14. The rule of falfe has been found very lifefui in finding 
approximates to the two unknown quantities contained in two 
given equations, and has been applied to («) equations having 
(») different unknown quantities: for example, it has been ob- 
ferved, that if two or more (m) values of an unknown quan¬ 
tity (*) are nearly equal to each other and to its given approxi¬ 
mate value (#'), the unknown quantity v = x - x' will afcend 
to two or more (m) dimenfions in one of the refulting equa¬ 
tions ; or in more than one equations will be contained fuch 
powers of the quantity (*u), that if the more equations were 
reduced to one whofe unknown quantity is v, the refulting 
equation will contain («) dimenfions of the quantity v. Hence 
it appears, that iii this cafe alfo the convergency of the ap¬ 
proximate. values found will depend on the given approximate 
being much more near to one root than to any other. 

j 5. When 



I'D $ 


Dr. Waring on 


15. When the given equations A = o, B = o, C = o, &c. 
contain irrational or other quantities whofe fluxions can be 
found ; and approximates ((0, h, c, d, &c.) are given to each of 
the unknown quantities (*, y, z, v, &c.) contained in the 
given equations; let a + x'— x, b-\-y' =y, c -f 2/ = z, cly-v' — v, 

&lc. : fubflitute in the quantities A,/ A ) 9 (—)> '(—■) > f A) » 

&c -* (I) • C|) • (i) - (D > *(i) • (J) - &c - 

x, y, %, u, &c. refpedively A, c, &c.; let the refulting cor- 
refpondeht values he A', jj^ A^, f —^ , &c.whence may be de¬ 


duced the equation A'+ ^A-) 4- (A)z' + ^^* j 


x y + f A:) #V + &c.) + &c. = o in which no irrational, &c. 

' XZ ' 

fun&ion of the approximates x\ y\ z\ &c. is contained; and 
•in the fame manner may the remaining equations B = o, C = q, 
&c. be transformed into others, in which no irrational func¬ 
tion of the approximates (x\ y\ z', &c.) is contained, and 
from the refulting equations may be found approximate values 
of the quantities x\y\ z\ &c. 

If there be given only two equations A=o and B = o con¬ 
taining two unknown qualities a? andy, and all the quantities of 
the refulting equations, in which are contained more than one 
dimenfion of the quantities x' and y' be rejected, there will re- 


fult the two .equations A' + f A) *'+/AT) y' = 0 and B'+] 

' # ' * j * 

(j)y “ °* ^ om ma r ^ found x' arid y\ the 

fame 



InfiniteSeries* 103- 

feme as given by Mr. Simpson andothers* When two or 
more values of the quantity x are nearly =<j, then in a refult- 
mg equation or equations, quantities of two or more dimen~ 
Sons of the approximate x' are to be included* 


PART lit 

1. THE firft algebraifts divided' quantities*- and extra&bd 
fheir roots no further than the quantities themfelves: they 
did not perceive the utility of proceeding any further^ other- 
wife the operation would: have been the fame continued* Mr*. 
Gregory St. Vincent, and Mr. Mercator divided, and 
Sir Isaac Newton divided and ex traded the roots of quan¬ 
tities (in which only one unknown quantity (x) is contained) 
by the operations then ufed by arithmeticians, into feries 
afcending or descending, according to the dimenfions of x in 
infinitum.. They clearly faw the utility of it in finding the 
fluents of fluxions, as- Dr. Wallis and others feme little 

m 

time before had'found the fluent of the fluxion ax” x-, or, which 

m 

is the fame, the area of a curve whofe ordinate is ax” and ah* 
feifla is x.. 

2. Mi. JLeibnitz: afked from Mr. Newton the cafes iiv 
which the above-mentioned feriefes would converge; for it 
would be altogether ufelefs when it diverges, and of little ufe 
when it converges flowly* 



je>4 I>. Waring on 

Caf. i. To this queflion an anfwer, I believe, was firft 
given in the Meditationes , viz. reduce the function to its loweft 
terms; and alfo in fuch a manner that the quantities contained 
in the numerator and denominator may have no denominator 
make the denominator Q = o, and every diftindt irrational quan¬ 
tity contained in it=o; and alfo every diftindt irrational quan¬ 
tity H contained in the numerator =o; then, let « be the leaft 
root affirmative or negative (but not = o) of the above-men¬ 
tioned refulting equations, the afcending feries will always 
converge, if the value of x is contained between « and — «j 
but if * be greater than a ,or - a, the above-mentioned feries 
will not converge. 

If the above-mentioned feries (S) be multiplied into x, and 
its fluent found; then will the feries denoting the fluent con¬ 
tained between two values a and b of the quantity (*) con¬ 
verge, when a and b are both contained between « and — a i 
the fluent always converges fafter than the feries S, the un¬ 
known quantity * having the fame values in both. 


—- = Ax + i 1 Cx 3 + &c. and 

t + bx-\- cx z + . . x" 

the roots of the equation a -\-bx cx -f- •. x n ~o be a, ( 2 t y, 


Ex. Let / - 

%J a ■ 


&c.; then 


a -f- bx 4* cx • • x 


f + JfL + Sec.=Ax + Bxx + 


G* ,I x + &c.; but 


W 

a—X 




^ -j- + &c. in infinitum , = 

ot, a otr P — X 


X_ 4. If + tf! + &c. &c. ; the former feries converges when v is 
ft & 

contained between a andthe latter when * is between 
/3 and / 3 , and fo on. In the fame manner the fluents 

■flL- Z x + ^+Scc.t r -iL = ^ + ^ + &c - &c. a fortiori eon- 

verge when x is between a and — «, @ and — See. refpec- 
twely, and fo on : hence the feries Ax + ! B* 1 + jC* 3 -f &c. 

= / 



Infinite Series. 


io 5 


-P—i --=/'—+ f-^ + f—+ &c. =fi+ | +, 

^ + &c.) x + ^jj ^ + &c.) a; 1 + &c. always converges 

when x is between a and—a, where « is the leaft root of 
the above-mentioned equation; but where x is greater than at 
or - u, the feries will diverge. 


The infinite feries a m + ma m ~' l x 4 - m . ——- a m ~ 1 x i 4- &c. — a+x” 

2 

will always converge when a is greater than x, and diverge when 
lefs, and confequently its convergency does not depend on the 
index m, unlefs when x — ^a: and in the Meditationes Analy¬ 
tic a are given the cafes in which it converges or diverges when 


■=pa = x; and alio if the feries x m -f- max'”~ J + &c. —x + a defcends 
according to the dimenfions of x, when it converges or diverges. 

Caf. 2. Let the above-mentioned quantity be reduced into a 
feries Ax ~' r + B« -r ~ ! -f&c. defcending according to the dimen¬ 
fions of the unknown quantity x ; then will the feries Ax~~ r -f 

B.v~ r ~*+ &c. =P, or the feries —-- - —— +&c. =z f Px con- 

l—r r J 

verge, when x is greater than the greateft root (x) of the 
above-mentioned equations, and diverge when it is lefs; and 
confequently in this cafe, when the fluent is required between 
the two values a and b of x ; the feries found will converge 
when a and b are both greater than X. 

Caf. 3. When x is equal to the leaft root in the former cafe, 
and to the greateft in the latter, then fometimes the feries will 
converge, and fometimes not. Thefe different cafes are given 
in the Meditationes; but it would be too long to infert them in 
this Paper. 

4. If any roots are impoflible as a — bs/ - 1 and a + b\/ — i,‘ 
then the feries will converge when x is in the firft cafe 
Vol. LXXVI. P lefs 



io6 Dr. Waring on 

lefs than dt= (a — b), and in the fecond cafe greater than 
=±=(<i+£); or, more general, it will converge in the firft 
cafe when x" is always infinitely lefis than the reciprocal 

1 • (0 + b V'Hi)»+/«_ £v'Z7) B = p , . . 

of the, quantity —— - ^ -, where n is infinite} 

(a*+6?) 

and in the latter cafe it will converge when x” is infinitely 
greater than P. 

It may not be improper to obferve, that the fame values of 
the root are to be applied in the equations, which are applied 
in the feties. 

3. Sir Isaac Newton, in the binomial theorem, reduced 
the power or root of a binomial into a feries proceeding ac¬ 
cording to the dimenfions of the terms contained in the bino¬ 
mial. M. de Moivre reduced the power or root of a multi¬ 
nomial into a like feries; but in all cafes, except the moft 
fimple, we muft dill recur to the common divifion, extradition; 
of roots, See. 

4. Meif. Euler, Maclaurin, and other mathematicians, 
finding that the feriefes before-mentioned often converged (lowly, 
or, if the truth may be con faffed, commonly not at all, to 
deduce the area of a curve contained between two values a and 
b of the abfeifs, of fluent of a fluxion between two values a 
and b of the variable quantity x, interpolated the feries or area 
between a and b ; that is, found the area or fluent contained be¬ 
tween the abfeiflae a and a + a, then between the abfeiffie a + ct 
and a + 2«, and then between the abfeifife a + and a + 3a, and 
fo on, till they came to the area between b — a. and b. M. 
Euler obferved, that when the ordinate became o or infinite, 
the feries exprelfing the area converges flowly; and therefore. 



Infinite Series. 107 

in order to inveftigate the area near the points of the abfcifs, 
where the ordinates become o or infinite, he transforms the 
equation, and finds feriefes exprefling the area near thofe points, 
in which feriefes the abfciflas or unknown quantities begin from 
thofe points. 

5. In the Meditationes it is afferted, that in a feries proceeding 
according to the dimenfions of ay if any root of the above- 
mentioned equations be fituated between the beginning of the 
abfcifs o and its end the feries wijl not converge : it is there¬ 
fore neceffary to transform the abfcifs fo that it may begin or 
end at each of the roots of the above mentioned equations, and 
confequently where the ordinates become o or infinite, &c.; 
thofe cafes excepted where the ordinate becomes o, and its cor- 
refpondent abfciffa is a root of a rational function W of x 

without a denominator, and f WPx is equal to the given 
feries; and in general the abfciffae ought to begin from the 
above-mentioned points; for if they end there the feries will 
converge very flow, if at all. 

6. It is further afferted, that if a and the values of the 
abfciffae between which the area is required, be both more near 
to one root of the above-mentioned equations than to any other, 
and n feriefes are to be found, whofe fum exprefs the area con¬ 
tained between a and b ; then that the fum of the (n) feriefes 
may converge the fwifteft, the diftances of the beginnings of 
each of the n abfciffae from the adjacent root will form a 
geometrical progreffion. 

7. Mr. Craig found the fluent of any fluxion 0 f the for¬ 
mula (a + bx n + cx ltt 4- &c.)'”.r s “’x by a feries of the following 

P 2 kind 



ioS Dr. Waring on 

kind (<* + 3 x* -f cx ln + &c.) M + I x x 9 x (a-j- Qx**\-yx % ” -f &c. in infi¬ 
nitum). Sir Isaac Newton, by feriefes of the fame kind, 
found the fluents of fluxions of this formula (a + bx” + cx M + 
See .") 1 x (e + fx n +gx' tn ■+- &c.) m x &c. rf—'x ; the fame principle 
is extended fomewhat more general in the Meditationes. 

8 . Mr. John Bernoulli found the fluent of any fluxion 


/ 


nz~nz 


Xj ft, | ; 

2# 2 


■ &c. from the principles which Mr. 


Craig published for finding the fluents of fluxions involving 
fluents. 

In the Meditationes fomething is added of the convergency 
of thefe feries ; and alfo. 

In them a new method is given of finding approxi¬ 
mations. Let feme terms in the given quantity be much 
Jefs or greater than the reft ; then reduce the quantity into 
terms proceeding according to the dimenfions of the fmall'' 
quantities, or according to the reciprocals of the great quan¬ 
tities, and it is done. If the fluent of the quantity rebuk¬ 
ing is required, find it from the common methods, if poffible j 
but if not, reduce the terms not to be found into an infinite Se¬ 
ries, and then find approximate values to each of the terms, &c. 
Ex. i. Let R the radius, and A the arc of a circle whofe 
fine is S and cofine C, and hd=.e an arc of a circle which* 
does not much differ from the arc A, that is, let e be a 
very fmall quantity: then will the fine of the arc A=fc£ be 

S(i — - x L— 

^ 2 R* r 2 


I e 3 

t 3 x r* + r 


3 • 4 


I 

-x - 

3. 4 R 4 2 

e 5 


X ^ + &c.) 


:C ( 


3 . 4 . 5.6 R s 
— &c.) ; and the cofine of the fame arc 


willbe C (i - — x L _p 


4 Rf 


• &c. .in. infinitum 



X' 


X 


% . 3 r 3 *2»3*4*S R s 

Ex. 2. Log. (x~*=.e) 

Ss-ix+ef -f: 


Infinite Series, 
&c.). 


A? , 

x±-e 


:log.Xd2- r zsz~±: e 


3 * 


I Of} 


'See,: 


, + &c. 


{x+e) z 

Ex. 3. /%-f— Tr z -f~2~r~~i 4 " See,.. 

J kJ a + (x — e) »/ a+x a + x 

Ex. 4. f -rr -4 -t*\ ~ f -JU=!=-- 4 =r=i=&C; 

v'Cl-O'i*) ) 

Ex. 5. r i 




/^ 7 +&c - 




f ^ i — jy 2 V i ..y 

fskrf^S: 




(1-/)**. 


Ex. 6. Let the fluxion be 


V"j~ 


-cx x 


V] 


, where c is a very fmall 


quantity; then, if P be put for s/i—x z , the fluxion becomes 

&c. of which the fluents will be found 


x cx % x C z x*x C 3 x b x 


2P 


8P ~ x6P 


A- 


, 1 . A-jrP A "?B x 3 ? c 3 „ 5C-« S P ,, , A 

_ x -- x 1 -- x 5 — - &c. where Arss. 

o 2 o 4 10 O 

fL, B=i^-,C=S=^, &c. 

This is the fwifteft converging feries for finding the length 
of the arc of an ellipfe nearly circular, which is yet known ; 
for example, let the abfeifs to the axis beginning from the 
center = x , the femi-tranfverfe axis of the ellipfe be 1, 
its femheonjugate 1 -d\ then will c— 2 d-d z , and let the 
length of the quadrant of the ellipfe be required, in this cafe 

x— 1, and P — s/1 —x 2 — o; and A as '^ T4 1 ^ 9 -—- = 1,57079, 
&c. whence the length required is 1*57079, &Ci x (i 


Qt a 



’ll o 


Dr. Waring on 


Ex.7. Let the fluxion be x.(a 3 4- a- -f + b ,s )x «f (3^ + r')v 2 

+ (/f + d ') # 3 ')’*= : Ptf, where a', b\ c' , &c. are very fmall quan- 

titles; then will Px =» ((a 4- bx)* m + ma -}- bx x (a' 4- b'x + c'x % 

4- d'x 3 ) + m. a^bx x (V 4- b'x + c'x 1 + d'x 3 y -f See.") x. 


of which the fluent is -—~—7 a ■\-bx‘ 4- m (a +bx) vn ~' i x 

3 m -t i . b s J 

/ d! , , f'— 3^A „ b'—2a3 . a 9 — aC\ . 0 , * 


(——-— a ; 3 + -—a ? 2 + - —a? + 7 —--r, ) + &c. where the 

letters A, B, C, &c. denote the preceding co-efficients. 

10. M. Euler and others, reduced the feries Av r +B^ r d-’4- 


Gv r + 2 ’ + &c. into a feries A' fln. r« + B fln. r 4- su + See. &c. 
where a, denotes the arc of a circle, whole fine is ax, &c. 
It may be eafily reduced into infinite other feriefes proceeding 
according to the dimen lions of quantities, which are functions 
of x ; but it is moll: commonly preferable to reduce it into 
feriefes proceeding according to the fines, cofines, tangents, or 
fecants of the arcs of circle, which fines, &c. Can immediately 
be procured from the common tables. 

It has been obferved in the firll part, that to find the root of 
an equation, an approximate value much more near to one root 
of the equation than to any other mull be given. In this part 
it is further obferved, that feriefes deduced from expanding 
given quantities, fo as to proceed according to the dimenfions 
of the unknown or variable quantities, wild not converge if the 
unknown quantities be greater than the leaft roots of the above- 
mentioned equations; and that they will not converge much, 
unlefs the unknown quantities have a fmall proportion to the 
leaft roots: and if the given quantities be expanded into 
feriefes defeending according to the dimenfions of the unknown 
quantities, then the feriefes refulting will not converge if the 

greateft 



III 


Infinite Series . 

greateft roots of the equations before-mentioned be greater than 
the unknown quantities; and unlefs the unknown quantities 
have a great ratio to the greateft roots the feriefes will converge 

flowly: for example,-the feriefesy^ x— 

— %— —&c., r~2= -y + 4y 3 -h&c. will never con- 

V i _ y 7, 

verge if x, z, ory, be greater than i ; but will always con¬ 
verge when lefsthan =±=i or ±rV — i theleaft or only roots of 
the equations/* -fv rrb, : i -f = o, and i -j-z 2 = o. The feries 
y+4y 3 + &c. will always converge when y is fituated between 
+ i and — i, in which cafe alone it is poffible. The fame is 
true alfo of a feries arifing from expanding t\\e j'(ax m + bx m ~ 1 + 

cx m -* -f &cc.fi m x into a feries proceeding according to the dimen¬ 
fions of if the equation ax m + bx m ~ 1 ■+• cx m ~ 1 + &c. = o has 
only two poffible roots a and — a, which are lels in the manner 
before-mentioned than any impoffible root contained in it. 

If in either of the above-mentioned leriefes the unknown 
quantity #, z , or y , has a great proportion to i, the 
feries will converge very flow; for example, if xzz-\ t ten 
thoufand numbers at leaft are to be calculated, to procure 
the fum of the feries true to four figures; therefore, in thefe 
and moft other feriefes it is neceflary firft to find a near value,. 
viz. when x either ~ z, when e is very fmall; . or — e, when z is 
very fmall ; and then write z + e for * in the quantity, and' 
reduce it in the former cafe into a feries proceeding according 
to the dimenfions of e, in the latter cafe according to 
the dimenfions of z, and; there will arife two feriefes, of 
which the fluents properly corrected, viz. by adding the 
fluent contained between the values a and c to the latter, 
and that between a and z to the former, will give the fame- 

fluent. - 



112 Dr . Wasi^g on 

fluent. ‘The firft term of the feries, in which e is fuppofed 
very fmall, will be the fluent of the given fluxion, when 
x = z. 

11. If a fluxion Pi, where P is a function of x, be transformed 
into another Q£, where Q_Js a function of 2, and they be 
reduced into feriefes A and B, proceeding according to the di- 
menfions of x and 2 refpedtively; find « and 7r, correfpondent 
values of the quantities x and 2; then in afcending feriefes, if 
a has a lefs ratio to the lead: root of the equation P = o, than 
has to the lead root of the equation Q=o, the feries A 
(exceptis exclpiendis) will converge fwifterthan the feries B. 

12. Dr. Barrow, in fome equations, exprefling the relation 
between the abfcifs x and ordinate jy, found y in the two fird 
terms of x , viz .y — a + bx, which is an equation to the afymp- 
totes of the curves. Sir Isaac Newton, from an algebraical 
equation given, exprefling the relation between y and x , found 
a feries proceeding according to the dimenfions of x , exprefling 
y in terms of x. M, Leibnitz performed the fame problem 
by afluming a feries Av"-|-Bx' , +'-f-Cv’'+ 2r -p&c. with general 
co-efficients, and fubdituting this feries for y in the given equa¬ 
tion, &c, from equating the correfpondent terms he deduced 
the indexes and co-efficients. M. de Moivre, Mr. Mac 
Laurin, &c. obferved, that when the highed terms of the 
given equations have two or more ( m ) divifors equal; for 
example, {y — ax") m ; to which we mud add, and when a 
value of y in this cafe is required nearly equal to Ax% a feries 

»+- 

Av B ri-B^ “ -f- &c. is to be afl’umed, whofe indexes differ 
only by - , &c. if otherwife they would differ by r. 

This reduction feldom anfwers any other purpofe than find¬ 
ing the fluents of fluxions as J'yx, &c.; or the afymptotes, &c. 

of 



Infinite Series. $Ij 

®f curves, which depend on fome of the firft terms of the 
feries ; but will very feldom be ufed for finding the roots of an 
equation ; the rule of falfe, or method given by Viet a, will 
ever be fubftitutedin its Head. 

13. The values of x may be required between which the above- 
mentioned feries A#"-hBtf”+ r -hC# B ^ 2r +&c. will converge, as 
the infinite feries anfwers no purpofe when it diverges. 'Firft, 
if an afcending be required, write for y in the given algebraical 
equation an infinite quantity, and find the roots of x in the 
equation thence refulting P ~ 0 ; which for y write in the fame 
equation, and find the roots of x in the refulting equation 
which contain irrational quantities, viz. if one root be x — a ; 
then let it contain (x — a') m , where m is not a whole number ; 
find the roots of the equations refulting from making every 
irrational fun&ion of (v) contained in the given equation ~ o, 
there being no irrational fun&ion of y contained in it; then, 
if x be greater than the leaft root not =0 of the above-men¬ 
tioned equations, the feries will not converge; but if it be a 
feries defcending according to the dimen fions of x, and x be 
lefs than the greateft root of the above-mentioned equations, 
the feries will not converge. 

In interpolating feriefes to inveftigate the fluent contained 
between two values a and b of the fluxion (A#"-f-B**+ r -h&c.)/x, 
it is preferable to make the abfciffae, begin from every one of 
the above-mentioned roots contained between a and b. 

Moft commonly thefe feriefes will not converge unlefs x 
be lefs, &c. than other quantities not inveftigated by this rule. 

14. Sir Isaac Newton gave an elegant example of this rule 
in the reverfion of the feries, y ** ax -f- bx 1 -j- of -|- &c. from which 
the inveftigation of the law of the feries has never been 
attempted. In the year *757 I fent the firft edition of my 

Vol, LXXVL Q Meditationes 



514 Br. Waring o# 

Meditationes Algebraica to the Royal Society, and publiffied it ins 
1760, and afterwards in 1762, with another part added, on the 
Properties of curves, under the title of Mifcellama Analytica, in 
all which was given the law of a feries for finding the fum of 
the powers of the roots of an equation from its co-efficients.. That 
great mathematician M. Le Grange and myfelf printed about 
the fame time an obfervation known to me at the time that; I 
printed the above-mentioned book, that the law of its powers 
and roots, if it proceeds in infinitum, is the fame; from 
which feries of mine, with great fagacity, M. Le Grange 
found the law which Sir Isaac Newton’s reverfion of feries 
obferves. In the Meditationes the law is given,, and the feries 
is made to proceed according to the dimenfions of x , &c. 

15. It is afferted in the Meditationes , that in moft equations 

of high dimenfions, unlefs purpofedly conftituted, the fum of 
the terms which-, from the given hypothefis, become the 
greateft, being fuppofed = o, only an approximate to the value 
Ax” of y in the refulting equation can by the common algebra 
be deduced. In this cafe the approximate to the quantity. A is 
to be found fo near as the approximate value of the quantity 
fought requires ; or perhaps it is preferable to corxefij: in. every 
operation the approximate values of the quantities A, R, C, &c. 
in the feries required- AV'-f + C / a”+ lr + &c. 

In the equation the quantity %-±ze may be fubftitutxd for x, 
and from the equation refulting a feries expreffing the value ofy 
may be found, proceeding either according to the dimenfions of 
the quantity or its reciprocals, according to the conditions of 
the problem, 

16. If there are more than one (») equations having («+i) 
unknown quantities (x,y, %, &c.) : in each of the equations for.y,. 



Infinite Series. 11 $ 

z, Sec. write refpe&ively Ax”, A'x m , Sec. ; and fuppofe the 
terms of each of the equations, which refult the greateft from 
the given or affumed hypothefis =o, and from the refulting 
equations may be found the firft approximates Ax*, A'x m , See. 
either accurately or nearly; then, in the given equations for 
y, z, Sec. write y' + (A + a) x* + Bx* +n ' + Sec.z' + (A' + a'') x m 
+ B / x m + m ’, where a, a?, &c. are very fmall quantities ; and fuppofe 
the terms of each of the equations which become greateft from 
the above-mentioned hypothefis refpeftively = o, and from the 
equations refulting deduce the quantities a, a'. See. ; n f , m!, 
See. ; B, B', &c.; and fo on : or affume y = (A+ la + ai -f 
&c.) x* + (B + ib+bi + &c.) x n + ni +&c.; z= (A' + ia' + a'i + 
&c.)x” + (B' + ib' + b'l + &c.) x m + m + &c. See. ; fubftitute thefe 
quantities for their values in the given equations, and from 
equating the correfpondent terms of the refulting equations 
may be deduced the quantities required. 

The differences of the indexes n', Sec. m', Sec. may be de¬ 
duced by writing x”, x m . Sec. for y, z. Sec. in the given equations, 
from the differences of the indexes of the quantities refulting; 
The fame principles may be applied in finding the above- 
mentioned differences, when two or more values are Ax”, Sec. 
which were applied in a like cafe to one equation having two 
unknown quantities. 

The fame principles which difeover the cafes in which a 
feries deduced from an equation having two unknown quanti¬ 
ties will converge, may be applied for the fame purpofe to 
thefe feries. 

17. In the equations for x,y, ss, &c. write refpe&ively x f + e, 
y'+f, z'+g - , &c.; and from the equations refulting find y', 
sf, &e„ in feriefes either proceeding according to the dimenfions 

Qj2 of 



is6 Dr. W aring on 

of tjie quantities e,f, g, &e.; or accordtitg to the dimenfions 
of the quantity x', as the conditions of the problem require. 

18. Given one or more («) algebraical equations involving 

unknown quantities, one unknown quantity (jy) may 
be exprefled by a feries proceeding according to the dimenfions 
©f the m— i remaining ones (x ., %, v. See.), in which any dimen- 
lions of 2, v, &e. aflumed may be fuppofed to correfpond to (/) 
dimenfions of the quantity (V). 

19. In a fluxional equation of the m xh order, expreffing the 
relation between y, and their fluxions, where x is conilant, 
Mr. Euler fubftitutes in the given equation for y m , y m ~\ y m ~\ 

Sec. refpedtively Ax"~ m x m r - — x m '~ l -p ax m ~~ l f : 
--- x n-m+z£»-z 4 -aXX”- 2 + bx m ~\ r ~—~^ 7 -— 

- --—- x m ~~ 3 3fr iax l x m ~3 +bxx m ~ 3 ■j‘ Cx M ~ 3 , Sec. where 

a, b, c. See. are any quantities to be aflumed in fuch a manner as 
the conditions of the problem require; from fuppofing the 
aggregate of the terms of the refulting equation, which are the 
greateft,=o,may be deduced the firfl: approximate A*”, or elfe (as 
is beforementioned) A'x” a near approximate to Ax”, and by pro¬ 
ceeding as in algebraical equations another approximate may be 
found, and fo on. The fame may be found by afluming y ~ 
Ax” -p Bx”+ r + CV+ 2 ’" + Sec. -p ax* -p bx m ~' -P cx m ~ % -p &c. or y 
(A+ \a-\-ai -p &c.)#’'-p(B + ib+bi -p&c.) x”+ r + (C-P ic-yci 
-p &c.)*"+* r -P &c. + ax’” -p bx m ~ 1 + cx”‘~ l -P &c. and fubftituting 

it and its fluxions for their values y, y, y, Sec. in the given - 
equation, and fuppofing the aggregate of each; correfpondent 
terms, which do not very much differ from each other, ~o; 
from the refulting equations can be deduced the co-efficients 
A, B, C, &c. ; or A, 1a, ai, See.; B, 1 b, b\ y Sec.; C, 1 c, ci, 
&c. &c. x In 



Infinite Series, I ij 

In the given equation for y, x, and their fluxions fubftitute 
/+/, x'y g, and their fluxions, where the quantities^/' and g t 
&c. are aflumed, fo as to render the quantities / and x f , & c. 
very fmall. 

In finding the feries which exprefles the value of y in terms 
of x , there will always occur as many invariable quantities to 
be aflumed at will as is the order of the fluxional equation, 
provided the feries begins from its firft terms ; and to find them 
there will refult equations eafily reducible to homogeneous, 
fluxional equations, of which the orders do not exceed m». 




