396 Dr. hutton on Cubic Equations 
greater than Jql ', which is = 5 2 = 25; and the greateft of 
the roots is pofitive, becaufe q - 1 o is pofitive ; and the 
two lefs roots negative. 
17. The equation x 3 — gx = — 10 has the fame three 
real roots as the former, but with the contrary figns, the 
lign of the greateft root being now negative, becaufe 
q — — 10 is negative. 
18. But the equation x 3 + gx = ±10 has only one 
real root and two imaginary roots, becaufe p — 9 is pofi- 
tive; and the fign of the real root is + or - according 
as the fign of q or 1 o is + or - . 
1 9 . The equation x 3 + 6x = -±io has alfo two imagi- 
nary roots, and one real root, w'hich is + or - as it is 
+ 10 or - 1 o, for the fame reafon as before. 
2,0. The equation x 3 - 6 x =■ ± 1 o has alfo two imagi- 
nary roots, becaufe - ^ 3 = 2 3 = 8 is lefs than ~[q^ = 5 2 = 25* 
2 1 . But the equation x 3 — iax = ± 16 has all its roots 
real, becaufe |^] 3 = 4 s = 64 is = ~q^ — 8 2 = 64. 
22. And the equation x 3 + 12X = — 16 has only one 
real root, becaufe/) = + 1 2 is pofitive. 
23. Let us now confider the other properties and rela- 
tions of the roots arifing from certain alfumed relations 
between e and r, and from confidering e either as real, 
imaginary, or nothing, that is e 2 as pofitive, negative, or 
nothing. 
24. When 
