Infinite Series* 1 1 5 
z, Sec. write refpe&ively Ax n 9 A'x m 9 See. ; and fuppofe the 
terms of each of the equations, which refult the greateft from 
the given or affumed hypothefis =0, and froyn the refulting 
equations may be fouiad the firft approximates Ax n , A'x m 9 &c. 
either accurately or nearly ; then, in the given equations for 
y 9 2, &c. write y' 4 ( A 4- ei) x n + Bx n + n ’ 4* Sec.z' 4- (A / 4- a /y ) x m 
4- B / t v w *+ w ', 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' 9 See . ; m\ 
Sec.; B, B 7 , &c. ; and fo on: or affume y •- ( A 4 1 ^ 4 a 1 + 
&c.) /4(B+ \b + b\ 4-&C.) x”+ nl 4- See,,; 2 — (A 7 4 -' la' + a'i + 
&c.)#*4-(B 7 4- lb' + b'i 4-&c.)jc aB + , " I 4- &c. &c.; 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 th* indexes n' 9 See. m ' 9 See. may be de- 
duced by writing x n 9 x m 9 Sc c. for y 9 2, &c. 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 n 9 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 9 y 9 2, &c. write refpe&ively x r 4 e 9 
y'+f, %/ +g 9 Sec.; and from the equations refulting find y ' 9 
9 See. in feriefes either proceeding according to the dimenfions 
Q^2 ‘ of 
