856] 
421 
856. 
NOTE ON A CUBIC EQUATION. 
[From the Messenger of Mathematics, vol. xv. (1886), pp. 62—64.] 
Consider the cubic equation 
a? + 3 cx + d = 0; 
then effecting upon this the Tschirnhausen-Hermite transformation 
y = xT x + (a 2 + 2c) T. 2 , 
the resulting equation in y is 
and this will be 
if only 
y 3 + 3y (cTi 2 + dT x T 2 - c 2 T 2 ) 
+ dT 3 - 6c 2 T 2 T, - ZcclTfL? - (d 2 + 2c 3 ) T 3 = 0, 
y 3 + 3 cy + d = 0, 
c = cT x - + dT x T 2 — &T£, 
d = dT 3 - QcMfT, - 3cdT x T 2 - (d 2 + 2c 3 ) T 3 , 
equations which give 
(<d 2 + 4c 3 ) = (d 2 + 4c 3 ) {T 3 + 3c r T x Ti + dT 3 ) 2 , 
viz. assuming that d 2 + 4c 3 not = 0, this is 
1 = T 3 + 3 cTfT? + dT 3 . 
Hence the coefficients T 1} T 2 being such as to satisfy these relations, the equation 
in z is identical with the equation in x; or, what is the same thing, if a, /3, y are 
the roots of the equation in x, then we have between these roots the relations 
/3 = aT x + (a 2 + 2c) T 2 , 
y =/3T 1 + (/3 2 +2c)T. 2 , 
a = yT x + (y 2 + 2c) T 2 ,