82 Dr. Waring on 
ceflive value of K ; let L = AxBxCx &c. x E 1 ^ x F 1 * x &c. 
x p r x q' x r' x &c* — A 1 ” x B m x C lr x &c. x E x F x &c. x p x 
q x r x &c. where p\ q\ r\ &c. are irrational quantities and 
fucceffive values of p , £, r, &c. The fadtors A B, C, &c. E 1 ^, 
F 1 *, &c. being given, the fadtors p\ q\ r', &c. into which they 
are multiplied in the quantity L will eafily be deduced by de- 
ducting the preceding irrational fadtors contained iti A, B, C, 
&c. E 1 *, F 1 *, &c. from the correfpondent irrational fadtors con- 
tained in L ; and in the fame manner, from the fadtors A 1 ”, 
B IW , &c. E» F, &c. can be deduced the irrational fadtors of the 
preceding />., q, r, &c. 
Aflume for the fum of the feries fought the quantity 
ExE / xE I 2 ,,E I ^ I xFxF,,F 1 ^ I x&c, 
A • A' . A ia . . A^xBxB'xB 1,4 . . B^-^CxC' . . C 1 • r - I x&c. 
x p x q x r x s x &c. {u% m ' -f + yz m '~~ 2 * 4 5 6 + &c.) -V ; where 
m r is a whole number, and ( 3 , y, &c. are co-efficients to be 
inveftigated ; write in V for 2 its fucceffive value 2+1, and 
let the refult be W ; reduce the difference W — V into a fradtion 
in its lowed: terms, and make the co-efficients of the correlpon- 
dent terms of the refulting fradtion = (W - V) and of the given 
fradtion equal to each other, if poffible ; 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 fundtion of % ; if not, it will proceed 
in infinitum , and may be exprefled either by a feries afcending or 
defcending according to the dimenfions of z . 
5. If any fadtor, A or B, or C, &c. have no fucceffive one in 
the denominator ; or if the greateft dimenfions of z in the de- 
nominator be greater than its greateft dimenfions in the nume- 
rator by 1 , then the fum of the feries is not a finite algebraical 
fundtion of 2. 
6. If 
