300 Mr. J. J. Sylvester on Aronhold's Invariants. 



f^-\-9'l/ for i/,he made to take the form fi:=x*-{-i/-\-6mx^y'^. 

 Then by the characteristic property of invariants, if I {a, b, c, d, e) 

 be any invariant to F of the degree q, we must have 



1(1, 0, m, 0, 1) = {f^-fgf' l[a, b, c, d, e) ; 



and it will be sufficient to prove that 1(1, 0, tw, 0, 1), or say more 

 simply I(w), can only have the two radically distinct forms cor- 

 responding to s and ty i. e, 



(5) = 1— 3»i* and (^)=m— m^, 



any other admissible form of I being a rational explicit function 

 of these two. 



It may be shown* that the parameter m in /will have six dif- 

 ferent values and no more. In the first place, if we write ix for 

 X (a meaning »/ —1), it is obvious that m becomes — m. Again, 

 let x + iy and x — iyhe substituted in place of a? and y respect- 

 ively ; then calling (/) the value assumed by F, when this substi- 

 tution is made, 



(/) = (^ + ,y)4+ (a?-,y)4 + 6m(^2 ^2/2)2 

 , = (2 + 6m) {x^ + 2/4) ^ ( _ 12 + t2m)xy 



= (2 + 6m){^^+/H-6-^^a.v}, 



Hence if we write 



X -\ ^ y for X, 



(2 4- 6m)* (2 + 6m) 

 and 



rjQ . y for y, 



(2 + 6m)* (2H-6m)*^ ^' 



and call what/^ becomes after these substitutions /g, 



/2={^^ + S^H67(m)^V}, 



7(m) denoting y^^;^^. 



In like manner, by writing in/g 



and 



1 . ^ f 

 — —. a?4- -r y tor 0? 



we obtain 



1 L f. 



rr X Ti- y tor y, 



/3=a^ + y« + 67V)A*» 

 * See addendum. 



