An introduction to the modern theory of equations, by Florian Cajori.

102

THEORY OF EQUATIONS

where the degree of the function H(x) does not exceed n - 1.
Now write a, for x. Since f(,,,) = 0, we have G (a,) = H(am),
and the theorem is proved.
80. The Tschirnhausen Transformation. The most general
rational algebraic transformation of a root of the equation
f(x) == 0 can therefore be represented by the integral functions
of the (n - l)th degree
y = c1, + dx + d312 + - * + Cdx'-1.
This is known as the Tschirnhausen transformation.
By its aid Tschirnhausen succeeded in reducing the general
cubic and quartic equations to the form of binomial equations.
We shall do this for the cubic,
box3 + 3 blx2 + 3b2 b 3 = 0.
We assume y = di + d2:V + x2, where dc and d2 are coefficients
whose values must be determined.
Let the roots of the given equation be ac, a2, at, and the
corresponding roots of the required equation y3 - c = 0 be
Af, WPf, o2P, where o and w2 are the complex cube roots of unity.
Then
/3 = c1 + da4 + a12, ]
11 = d 4- (-da, + a2, I
W)2 = C(l + d(1.a3 + C3.
Adding, we obtain 3 di + d.,s, - s =- 0.
Multiplying the second equation by w, and the third by o2, and
adding, we have (a1 +- oj a + aa3) 2 + 2a2(2 = 0.
Whence
d2 + S1 = d2 + a1 + 2 a3 - -9a3 + a, +. 1
(1 + Wa"2 + W a3
Since w may represent either one of the two complex cube
roots of unity, there are two possible values for this fraction.