Mr. J. J. Sylvester on Aronhold's Invariants, 301 



where ~ 1 



-—1 + 



2 l+3m _ — 2— 2m _ —1— m ^ 



7(?w) is a periodic function of m of the third order^ for we find 



3/ N ^f \ — (l + 3m)--( — 1+m) 



It will of course be observed,, also, that 



rfm=.—(y[—m) and ym=~-y'^{ — m). 

 Hence 



( — 7)(— 7)m=— 7^(— ??2)=m( — 7^)(— 7^)m= -— 7^( — m) =m. 

 So that^ in fact, the six values of the parameter are 

 m, ym, y^m 

 — m, —7m, — 7^m, 

 forming two cycles, having the remarkable property that the 

 terms in the same cycle are periodic functions of the third order 

 of one another, and each term in one cycle is a periodic function 

 of the second order of every term in the other cycle. 



The modulus of substitution for passing from /^ to f^ i, e. the 

 square of the determinant 



1 L 



(2 + 6m)^' {2 + 6mf 

 1 -i 



( -20^ . -2 



is (2-\-QmY ^'^' l + 3m* 



(2 + 6m)^' (2 + 6w)*' 

 So that if I(m) be the value of any invariant of the degree q, 



corresponding to the form /j, and consequently I ( :j — ^ j the 

 same for /g, we must have -t-om/ 





In like manner, by means of /g it may be shown that we must 

 have the further equation 



I^=(^__^j.l(^j_^j. 



These equations are easily verified for the values of {s) and {t). 

 Thus 



_ _ (l-3m)^ / m + l _ / m + iyi . 

 "" 8 Ll~3m \- 3w/ J ' 



