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



and it is moreover obvious, that the values of (s) and {t) might 

 have been found a priori by means of these functional equations. 

 The essential point of inference for my present purpose from 

 the equations above which are of the form 



•-■ I'»=Hxl(^) = Kxl(SL) 



is this, that if I(m) contain any power of m, say m*, it must 

 also contain (m — 1)' and (m+ 1)*; in a word, {m^—my, which, by 

 the way, it may be noticed is f. Now, if possible, let there be 

 any invariant I (m) of the ^th degree in (m) which is not a 

 rational function of (s) and {i). If we make 2x-\-Sy=q, as 

 many integer solutions as exist of this equation (in which zero 

 values of x and y are admissible), so many functions of the form 

 (s)* . (ty may be formed of the degree (q) in (m), and all of 

 them of course invariantive functions. 



As regards the general nature of any invariantive function in 

 (m), since the change of x into -r-xin x^ -\-y^-\- ^mx^y^ introduces 

 no change into the invariant if q be even, but changes the sign 

 if 5' be odd, it follows that I^(7w) is of the form ^{mf when q is 

 even, and of the form m<j>{m^) when q is odd. 



Let fi be the number of solutions of the equation in integers 

 above written. Then, by linearly combining all the different 

 values of {s)", {ty with Iq{m), it is obvious that we may form a 

 new invariant, say.T^, in which the fju first occurring powers of 

 m will be wanting, i. e. in which the indices 0, 2, 4, ... (2/>6— 2) 

 will be wanting when q is even, and i, 3, 5, ... 2/x — 1 when q is 

 odd. Hence in the former case the new invariant will contain 

 m^, and in the latter case w?^'^^ -, and therefore, by virtue of 

 what has been shown already, I'^ will contain {m^—myf^ in the 

 one case and (m^— 7w)V+i in the other. 



1. Let q = 6i, or 62 + 2, or 6i + 4; then/x = i + l ; and there- 

 fore [m^—mf^'^^y which is of the degree ^2 + 6 in (m), is contained 

 as a factor in I which is of the degree {q) only, a quantity less 

 than 6i-f 6, which is absurd. 



Again, 2nd. Let §' = 61 + 1, then /a=2 ; and (yw^— m)V+i is 

 of the degree 6z + 3 in (m), and is contained as a factor in I, 

 which is of the degree Gz + l, which is again absurd. 



So, 3rd. If ^ = 6j + 5, ft=iH-l; and the factor (m^— m)2/i+i 

 is of the degree 61 -f 9, which is still more absurd. 



Finally, if g'=6i + 3, /i=i + l; and the factor becomes 

 (m^— Tw)^"*"^, i. e. (m^—mY, and consequently the entire value of 

 I' is (/)' ; but I differs only from 1' by linear combinations of 

 powers of («) and (/). Hence I on this' supposition cannot be 



