488 



t. xlvi. p. 961 (1858), contains a very important theorem in relation 

 toTschirnhausen's transformation of an equation/ (.r)=0 into another 

 of the same degree in y, by means of the substitution y ^x y where 

 $x is a rational and integral function of x. In fact, considering for 

 greater simplicity a quartic equation, 



(a,b,c,d,ejx t iy=0, 

 M. Hermite gives to the equation y=fyx the following form, 



(I write B, C, D in the place of his T , T\, T 2 ), and he shows that 

 the transformed equation in y has the following property : viz., every 

 function of the coefficients which, expressed as a funtion of 0, b, c, d, e, 

 T, B, C, D, does not contain T, is an invariant, that is, an invariant 

 of the two quantics 



(, b } c, d, eJX, Y) 4 , (B, C, D Y,-X) 2 . 



This comes to saying that if T be so determined that in the equation 

 for y the coefficient of the second term (y 3 ) 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 



and we have thence for the value of y, 



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-mentiond expression for y, 

 all the coefficients except those of the term in #, or 6B + 3eC + 3rfD 

 are those of the binomial (1, I) 4 , whereas the excepted coefficients 

 are those of the binomial (J, I) 3 ; this suffices to show what the ex- 

 pression for y is in the general case. 



I have in the two papers, " Note sur la transformation de Tschirn- 

 hausen" and "Deuxieme Note sur la transformation de Tschirn- 

 hausen" (Crelle, t. Iviii. pp. 259 and 263, 1861), obtained the trans- 

 formed equations for the cubic and quartic equations ; and by means 

 of a grant from the Government Grant Fund, I have been enabled 

 to procure the calculation by Messrs. Davis and Otter, under my 



