ON TSCHIRNHAUSEN’S TRANSFORMATION. 
[From tho Philosophical Transactions of the Royal Society of London, vol. cli. (for the 
year 1861), pp. 561—578. Received November 7,—Read December 5, 1861.] 
The memoir of M. Hermite, “ Sur quelques théorèmes d’algèbre et la résolution 
de l’équation du quatrième degré,” Comptes Rendus, t. xlvi. p. 961 (1858), contains 
a very important theorem in relation to Tschirnhausen’s Transformation of an equation 
f(x) — 0 into another of the same degree in y, by means of the substitution у = фх, 
where фх is a rational and integral function of x. In fact, considering for greater sim 
plicity a quartic equation, 
(a, b, c, d, e^x, 1 ) 4 = 0, 
M. Hermite gives to the equation у = фх the following form, 
у = aT + {ax + 45) В + (ax 2 + 4bx + 6c) C + (aa? + 4bx 1 + Qcx + 4<d) D, 
(I write B, C, D in the place of his T 0 , T 1} T 2 ), and he shows that the transformed 
equation in у has the following property : viz., every function of the coefficients which, 
expressed as a function of a, b, c, d, e, T, B, C, D, does not contain T, is an invariant, 
that is, an invariant of the two quantics 
(a, b, c, d, e\X, Y)\ (.B, C, D&Y, - X)\ 
This comes to saying that if T be so determined that in the equation for у the 
coefficient of the second term (y 3 ) shall vanish, the other coefficients will be invariants ; 
or if, in the function of у which is equated to zero, we consider у as an absolute 
constant, the function of у will be an invariant of the two quantics. It is easy to 
find the value of T ; this is in fact given by the equation 
0 = aT + SbB + 3cC + dD ;