be of even tiiMteiS*^^ fcV'Hf ^^ ^grie^ of the dimensions be 

 (q), and D the determinant of the coefficients of substitution, 

 the invariant to the transform becomes the original invariant 



affected with- a factor Dp", where — must be an even integer, 



P . . 

 since otherwise the sign of this multipher would be equivocal and 



indeterminable ; hence when i and p are both odd, q must be 

 even. Thus, then, l{m) in the case before us must be an even- 

 degreed function of m. Moreover, since the change of x into px 

 converts minto pm, and Iqim) into p^{m) [for D becomes p when 

 x, y, z become px, y, z], lq{m) must be of the form (f>{m^), 

 m^<f>{mY, m^^{mY, according as the index {q) is of the form 

 62, 6z-f2, 62 + 4. 



By precisely the same reasoning as was applied to the pre- 

 ceding case of {s) and {t), we see that any invariant of m which 

 contains m'- must also contain (1— m)^, {\ — pmY, {l—p^mYj 

 i. e. must contain {m—m^Yf which in fact is (S)'^. If, now, we 

 consider any invariant of the ^th degree in (m) I(m), and sup- 

 pose it to be other than a rational function of (S) and (T), 

 and if we take (//,) to denote the number of the solutions of 

 4:x-\-Qy=qy it will follow that we may form an invariant l'(m), 

 which, when q is of the form \2i or 122 + 6, will contain m, and 

 consequently [m—m^yf^'^^ as a factor; and in like manner when 

 q is of the form 122 + 3 or 122 + 8, will contain (m— m'*)¥+2 as 

 a factor j and when q is of the form 122 +,4 oj??122 + 10 will con- 

 tain {m—m!^Y^^^ as a factor. Now 



,f^}^ aqOiCv^ ^n^ ^y^i ^ = 2 + i 



n '" 



\u 



when 



^ = 122 + 6 /-tsssf+i; 



g = 122 + 2 ^6 = 2*,' 



5 = 122 + 8 ;t^ = 2 + l; . 

 when -l^^j^V 



6^=122 + 10 /^ = 2 + l-'V ' 



i .... (.. §' = 122 + 4 jW, = 2 + 1. 



Hence the factors dividing I^ in these several cases will be of the 

 respective degrees 



122 + 12, 122 + 12; 122 + 8, 122 + 12; 122 + 16, 122 + 16; 

 corresponding to {q), being of the several values 



122,122 + 6; 122 + 2,122 + 8; 122 + 10,122 + 4; 



which is clearly impossible. This proves the theorem in question 

 (the passage being made from the canonical to the general form, 

 as in the former part of this investigation), to wit, that S and T 

 form what I have elsewhere termed a fundamental scale of inva- 



