Second Cafe of the Cubic k Equation x'-qx—r. 921 
of this paper is to {hew how, by the help of Sir isaac 
newton’s famous binomial theorem, the foregoing fo- 
lution of the other, or firft, cafe of this equation may 
be, as it were, extended to this latter cafe, or, rather, 
may be made the means of difcovering, by a very pecu- 
liar train of reafoning, another folution, that {hall be 
adapted to it. 
2 1 . By the binomial theorem it appears that the cube- 
root of the binomial quantity a + b (in which a is fup- 
pofed to be greater than b ) is equal to the following in- 
finite feries, to wit, a> + - — 
7 7 5a 
ah 1 
lOaib^ ^ 22 aib s 
8itf 3 729 a % 
154 aib* ^ 261 Satb 7 
6561a 0 ' lyjrfftia 1 
_ lob * 
81 a i 
&c. or to cfi x the infinite feries i +— - 
2 2$ S 
zzv 1 5 <± b ' 2010 o fy /• r 
+ — 7 - + r - &x. or (if we 
720 a s 6 c 6 ia 6 K 
i6i8£ 7 
729 a * 6561a 137,78 
9 a~ Via* 243 ^ 
put the capital letters a, b, c, d, e, f, g, h, Sec. for the fe- 
veral numeral coefficients, 1 , h -rr, 1 
54 
2618 
43’ 7*9’ 65 6 1’ i 3 7, 78 1 ’ 
8cc. of the terms of the feries, refpe£tively,) a s x the 
infinite feries j iSl - 
xa 6 a qa 6 nor icr 18 a 21a' 
Sec. in which feries both the numerators and the deno- 
minators of the generating fractions, |, |, f, 
Sec. following the fecond term, increafe continually by 
3, fo that it will be eafy for any one to continue the fe- 
ries to as many terms as he ffiall think proper. 
22. In like manner the cube-root of the refidual 
quantity a-b is found by the fame binomial theorem to 
Vol. LX VIII. 5 Y be 
