so

ON CUBIC EQUATIONS

TRACT 28.

\ z = WH + Vld*!) 1 + (i/>) 3 ])x 1 or x -

(y = v'Ci 1 - VliiqY + (i/>) 3 ]) X 1 or x - \ ’

the three values of z and y ; for every quantity has three-different forms of the cube root, and the cube root of I, is

not only 1, but also --—— or-——. Hence then the

three values of z + y or x, or the three roots of the equa-tion x 3 -f px — q, are

l/(iq + \ZU\qY + {ipY}) x 1 or x - - + or x 1-

1/(19 -VUilY + (t PYl) x 1 ° 1- x °r x - 1 +

where the signs of must be opposite, in the values

of z and y, that is, when it is - 3 in the one, it must be

—- in the other, otherwise their product zy will not

be = — j-p, as it ought to be.

42. Or if tve put a zz ^p, and b — |</, the same threeroots will be

l/\b + ^{b l + « 3 )] + l/[b — \/(b'- + a 3 )] = the 1st root or r,

-iVib+</{&+a 3 )-]. (I -*/- 3)

— J -l/\b — ^/(b 2 4- a 3 )]. {1 + */ — 3) the 2d root.

— -z\/lb — */(b 2 + a 3 )]. (1 — — 3) the 3d root.

43. Or again, the 1st root r being

l/\b + x /{!> 2 + a 3 )] + l/[b — x /(b 2 + a 3 )], the other twowill be

- F + + « 3 )] F(/>M-.«*)] =

the 2d root, and

- f - ^ W + V(t> z + <* 3 )] + nr M* - V(b 2 +a*)] =

the 3d root.

44. Or, if we put s = %/\b + (b 2 + a 3 )], and d zs

S/\b — */{b 2 4* a 3 )], the roots will be