98 PROFESSOR CATLEY ON TSCHIRNHATJSEN'S TRANSFORMATION. 



The expression for the quadrinvariant was, as already mentioned, given in the former 

 memoir : I find that the two invariants are in fact the invariants of a certain linear 

 function of U, H; viz. the linear function is =U'U-|--|0'H; so that, denoting by I*, 

 J*, the quadrinvariant and the cubinvariant respectively of the form 



(1, 0, c, ©, eiy, ly, 



we have 



I*=T(U'U + 40'H), 



J*=J(U'U+40'H), 



where I, J signify the functional operations of forming the two invariants respectively. 

 The function (1, 0, C, JB, ^^, 1)S obtained by the application of Tschirnhausen's 

 transformation to the equation 



(a, b, c, d, eyx, 1)*=0, 



has thus the same invariants with the function 



U'U + 40'Hr=U'(a, b, c, d, eXx, l)^+40'(ac-J^ ad-bc, ae+2bd-Sc\ be-cd, ce- d'Jx, l)^ 



and it is consequently a linear transformation of the last-mentioned function ; so that 

 the application of Tschirnhausen's transformation to the equation U=0 gives an 

 equation linearly transformable into, and thus virtually equivalent to, the equation 



U'U+40'H=O, 

 which is an equation involving the single parameter -w : this appears to me a result of 



considerable interest. It is to be remarked that Tschirnhausen's transformation, 



wherein y is put equal to a rational and integral function of the order n—1 (if w be the 



order of the equation in x), is not really less general than the transformation wherein 



V 

 y is put equal to any rational function ^ whatever of x ; such rational function may, in 



fact, by means of the given equation in x, be reduced to a rational and integral function 



of the order n—1; hence in the present case, taking "V, W to be respectively of the 



order n—1, =3, it follows that the equation in y obtained by the elimination of ^ from 



the equations 



(a, b, c, d, ejx, 1)^=0, 



^ («', /3', y', S'X^, 1)« 



is a mere linear transformation of the equation AU+BH=0, where A, B are functions 

 (not as yet calculated) of {a, b, c, d, e, a, /3, y, 5, a', /3', y', S'). 



Article Nos. 1, 2, 3. — Investigation of the identical equation 

 JU"'-IU'^H'+4H"+M0'= -$'^ 

 1. It is only necessary to show that we have such an equation, M being an invariant, 



