200 Prof. H. J. S. Smith on Quadratic Forms [Dec. 5, 



e.-i 



I. If f^i^2, 5- has the character (— l) ^ 



ej-i 



II. If i^i^ 3, 0^, in addition to the character (—1) ^ , has also the cha- 



racter (— 



III. If fj(.=: I, and also ^ 2, .^j ^ 2, 0. (which, as well as and 

 is necessarily uneven) has the character 



(_1) 8 or(-l)~+"r- 



according as 



(_l)H^i-i-i) + K^e+i-i) = (_l)Ki;-+i), or =(_l)Hi'.-i). 

 It will be observed that (by I) the forms and 9^^^ have the charac- 



ters(-l) 2 and(-l) 2 . 



IV. If f^i = 0, and also 2, fi^-^j^^ 2, 9-, if uneven, has the charac- 



e;-i 



ter (— 1) ^ , or no character at all, according as 



(_l)H%-i-i)+KVi-i^::^^_l)l(i.-i) or (_l)Kii+i). 

 No even concomitant has any supplementary character. But if 5. is an 

 even concomitant, the uneven forms preceding and following it have, by I, 

 the characters 



(-1) 2 ,and(-l) 2 • 

 These characters are not independent but are connected by the equation 



(_l)H^i-l-l)+^ ^Vl-'^=:(_l)*^'' + ^>. 



Thus if I-, lij^i, . • • Ii+2j i'^ ^ sequence of an uneven number of uneven in- 

 variants preceded and followed by even invariants, and corresponding to a 

 sequence of alternately even and uneven concomitants Q., 9.^^ . . . 

 the character, mod 4, of every uneven form of this sequence, and of the next 

 following form 9^+2^+1' determined by the character of the form 

 We have, in fact, if s= 1, 2, . . .j+l, 



^_^y^'i-v2.-i ^^=(_iyx (-1) X (-1) 



Besides these supplementary characters, which, no less than the principal 

 characters, are attributable to individual forms of the concomitant system, 

 there exist, or may exist, other characters, which we shall term simulta- 

 neous, attributable to certain sequences of those forms considered conjointly. 

 Such a character is attributable to every sequence of uneven forms, of 

 which none possesses any sup]-)lementary character but which are imme- 

 diately preceded and followed by forms having such characters. The fol- 



