41 6 Dr. hutton on Cubic Equations 
Memoir es de V Acad. 1738. Alfo 
s-d = :£+ - 
■ds , 3 
liifL. x _ 2 • 5 • 8 « 1 k 4 Oj- c Therefore 
3.6.9.12.15^ 1 nererore » 
_ j+£ 
2 
=^— nZ -3 
2 J 
3 . 5 • 6 . 90* 
— \/ b x : 1 - 
2C 
- V 5 ’ 8 ‘ 4 / 4 
3.6.9. 12 £ 4 
3 -^ 
£ i 2 . Cc 2 2 . C . B . IIC 
x :l + T-r 2 m + — 
#b' 
3 3 . 6 . 9^“ 3.6.9.12. 15^ 
8cc, 
for the other two roots, which were given by clairaut, 
in his Elemens d'Algebre » 
69. Hence again it appears, that when c 1 is pofitive, 
thefe two latter roots are imaginary; for then the fa6tor 
SdCtJ i s imaginary. And that thofe roots are real when 
s/p 
this c' is negative; for then this factor becomes 
v/ ~ 3 = e -^k , a real quantity. But in this laft cafe, 
the lign of every fecond term in the two feries muft be 
changed, namely, the figns of the terms containing 
the odd powers of the negative quantity c 2 ; for the feries 
contain the letters as adapted to the pofitive fign only. 
70. Thefe feries are proper for thofe cafes only in 
which c 1 is not greater than A; for if c 2 were greater 
than b % they would all diverge, and be of no ufe : and 
the feries proper for the other cafes, namely, in which c\ 
is greater than b* f we fliall give below. 
*ji. That; 
5 
