542 Mr. Cayley on Tschirnhausen's Transformation. [Dec. 7, 



riants of the first, and of 9', =BD C 2 , the invariant of the second of 

 these two forms, viz. that we have 



<e=iu /3 -3H' 2 +Fe' 2 +i2j'e'U'+2re'H'. 



And by means of these I obtained an expression for the quadrinvariant of 

 the form (l, 0, (, J9, <E[jy, 1 ) 4 ; viz. this was found to be 



= KP + fl 2 0' 2 +12J0'U'. 



But I did not obtain an expression for the cubinvariant of the same func- 

 tion : such expression, it was remarked, would contain the square of the 

 invariant $' ; it was probable that there existed an identical equation, 

 JU' 3 IU' 2 H' + 4H' 3 +Me'= $' 2 , which would serve to express #' 2 in 

 terms of the other invariant ; but, assuming that such an equation ex- 

 isted, the form of the factor M remained to be ascertained ; and until this 

 was done, the expression for the cubinvariant could not be obtained in its 

 most simple form. I have recently verified the existence of the identical 

 equation just referred to, and have obtained the expression for the factor 

 M ; and with the assistance of this identical equation I have obtained the 

 expression for the cubinvariant of the form (l, 0, C, 3B, <^V, l) 4 . 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 + ^9'H; 

 so that, denoting by I*, J* the quadrinvariant and the cubinvariant re- 

 spectively of the form (l, 0, <, U, (jy ) 4 , we have 



I*=I(U'U + 49'H). 



J*=J(U'U + 49'H), 



where I, J signify the functional operations of forming the two inva- 

 riants respectively. The function (l, 0, (, 39, &][y, l) 4 , obtained 

 by the application of Tschirnhausen's transformation to the equation 

 (a, b, c, d, ej^x; l) 4 =0, has thus the same invariants with the function 



U'U + 40'H 



:U'(, b, c, d, e&x, l) 4 +49'(c-J 2 , ad be, ae + 1bd-3<?, be-cd, ce-drj^c, l) 4 , 

 and it is consequently a linear transformation of the last-mentioned func- 

 tion ; so that the application of Tschirnhausen's transformation to the 

 equation U=0 gives an equation linearly transformable into, and thus vir- 

 tually equivalent to, the equation U'U + 49'H=0, which is an equation 



involving the single parameter : this appears to me a result of con- 

 siderable interest. It is to be remarked that Tschirnhausen's transforma- 

 tion, wherein y is put equal to a rational and integral function of the order 



