4io D/v Hutton on Cubic Equations 
the faid two cafes ; and confequently s -~ \/- 3 = v/sfc 3 £ 
a real or an imaginary quantity according as the roots 
are to be real or imaginary. 
50. The firft root r being found from the formula 
\/b + V p + a* + \^b - ^ b % + a\ or by any other means, 
the other two roots may be exhibited in feveral other 
forms befides the foregoing, as may be lhewn in the fol- 
lowing manner. 
51. The equation being x 5 + px — q, and one root r, 
by fubftitution we have r % + pr - q, and, by. fub- 
tradting, it is - - - — r z + p . x — r — o, and, 
dividing by x - r, it becomes x* + rx + r* + p = o. 
Or this fame equation may be found by barely di- 
viding x* + px - q - o by x - r = o, for the quotient is 
x 1 + rx + r 1 + p = o. And the refolution of this qua- 
dratic equation gives x = - ± v / _ p - 
l >/— 4/> - ir 1 the other two roots. And from hence 
again it appears, that thefe two roots are always imagi- 
nary when p in the given equation is poiitive; as alfo 
when it is negative and lefs than | r 1 ; which again in- 
clude all the cafes of the table of forms after the 1 3th. 
52. Again, fince r 5 + pr = q, therefore r* + p = 
and r* = - p + ~ , and —3 r l = 3 p-j-i which being 
3 fubftituted 
