1 1 6 Dr . Waring m 
of the quantities e 9 f 9 g , &e. ; or according to the dimenfions 
of the quantity x\ as the conditions of the problem require. 
18 . Given one or more (;z) algebraical equations involving 
(n 4 rn) unknown quantities, one unknown quantity (jy) may 
be expreffed by a feries proceeding according to the dimenfions 
of the m — i remaining ones (x, % r v 9 &c.), in which any dimen- 
fions ofz, v, &c afi'umed may be fuppofed to correfpond to (/) 
dimenfions of the quantity ( x ). 
1 9 . In a fluxional equation of the m th order,, e^prefiing the 
relation between x 9 y 9 and their fluxions, where x is conftant, 
Mr. Euler fubftitutes in the given equation for y m 9 y m ~\ y m ~~ z -» 
&c. refpefHvely Ax n ~”> x m 9 - — — — x?* 1 * 4 
r J n — m + I 
A 
X n ~ M + 1 X rn ~~ 1 4 QXX m ~~ z 4 bx m ~ z 9 
1 ) {n — m-b 2) " ’’ 8 ” 7 1) (n-m-b2) 
7 — 3 x m ~~ 3 4 * * 4- 1 ax z x m '~~i 4- bxx m ~~* 4 cx m ~i. &c. where 
(u — m + 3 ) 
b 9 c, &c. 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 firft approximate A*%or elfe (as 
is beforementioned) A'x n a near approximate to A#% and by pro- 
ceeding as in algebraical equations another approximate may be 
found, and fo on. The fame may be found by affuming y = 
Ax' 1 4 B,v”+ r 4- Cx n + 2r 4 &c. 4 a x m 4 bx m ~~ l 4 cx m ~~ z 4 &c. or y » 
(A 4 la^ai 4 &c.) y” 4 (B 4 ib + bi 4&c.) #?+' + (C4 i^4ci 
4 &c.)v"+ 2r 4 &c. 4 4 bx vl ~^ 1 4 cx w ~~ l 4 &c. and fubflituting 
Stand its fluxions for their values &c. 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, ia y ai 3 &c>; B, 1 b y bi 9 &c, ; C, ic> ci 9 
&c. &e, 1 
In, 
