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XXIV. On Tschienhausen’s Transformation. By Arthue Cayley, Esq.., F.B.S. 
Eeceived November 7, — Eead December 5, 1861. 
The memoir of M. Heemite, “ Sur quelques theoremes d’algebre et la resolution de 
I’equation du quatrieme degre,”* contains a very important theorem in relation to 
Tschirnhausen’s Transformation of an equation /’(^)=0 into another of the same degree 
in y, by means of the substitution y—cpx, where (px is a rational and integral function 
of X. In fact, considering for greater simplicity a quartic equation, 
{a, h, c, d, ejx, 1)^=0, 
M. Heemite gives to the equation y=<^x the following form, 
yz=:aT-\-{ax-{-h)B-\-{ax^-\-ihx-\-Qc)C-\-{aaf-\-ihx'^-\-^cX’\-4:d)T) 
(I write B, C, D in the place of his To, Tj, Tg), and he shows that the transformed equa- 
tion in y has the following property : viz., every function of the coefficients which, 
expressed as a function of «, 5, c, d, e, T, B, C, D, does not contain T, is an invariant, 
that is, an invariant of the two quantics 
(«, 5, c, d, ^XX, Y)^ (B, C, DXY, -X)^ 
This comes to saying that if T be so determined that in the equation for y the coeffi- 
cient of the second term {y^) shall vanish, the other coefficients will be invariants ; or if, 
in the function of y which is equated to zero, we consider y as an absolute constant, the 
function of y 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=«T-l-35B-j-3cC-l-cZD; 
and we have thence for the value of y, 
y={ax-\-i>)^-{-{ax^-\- ^hx-\-2>c)C-\-{ax^ -{-Aihx^ -\-^cx-\-^d)J ) ; 
so that for this value of y the function of y which equated to zero gives the transformed 
equation will be an invariant of the two quantics. It is proper to notice that in the 
last-mentioned expression for y, all the coefficients except those of the term in x^, or 
JB-]-3cC-f-3(ZD, are those of the binomial (1, 1)^, whereas the excepted coefficients are 
those of the binomial (1, 1)^; this suffices to show what the expression for^ is in the 
general case. 
I have in the two papers, “Note sur la Transformation de TscHiR]srHAUSEH,”f and 
“Deuxieme Note sur la Transformation de TsCHiR]srHAUSE]V,”f obtained the transformed 
* Comptes Eendus, t. xlvi. p. 961 (1858). f Ceelm, t. Iviii. pp. 259 and 263 (1861). 
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