424 Dr. hutton on Cubic Equations 
4 /c x n/- — 3 will be imaginary. But if c z be negative, 
tliofe roots will be both real ; fince 4 / c x V —3 then be- 
comes 4 / c . %/— 1 x 4 /— 3 — 4 / c x - %/—i x 3 = 
- 44 c x v/3. The fig 11 s prefixed to the terms as above, 
take place when c z is pofitive ; but when c r fhall be ne- 
gative, the figns of the terms containing the odd powers 
of it; muff: be changed. And thefe feries include all the 
cafes in which the former ones failed by not converging. 
So that between them they comprehend all the cafes of 
the general cubic equation X 3 ±- px — q, as they each re- 
ciprocally converge when the other diverges, but in no 
other cafe, except in the common clafs, in which c is = b y 
which happens at the two limits, namely, either when 
a is = o, or when — a i = 2 b l : and then they both give 
the fame roots. But in the other cafes they give the 
contrary roots; namely, when c is lefs than b, the firft 
feries gives the greateft root; and when c is greater 
than b } the latter feries gives the leaf! root. 
93. Now when a is any pofitive quantity, the firft of 
thefe feries gives the only real root, without any change 
in the figns of the terms; the other two being imagi- 
naryi And this includes all the cafes after the 1 6th in 
1 , . , ' f 
the table in Art. 30. 
94. When a is = o, or the limit between pofitive and 
negative, as in the rbth form in Art, 30. then is c sc b, 
a and 
