1867.] Prof. H. J. S. Smith on Quadratic Forms ^c. 



201 



lowing definition is requisite in order to explain the nature of these simul- 

 taneous characters. 



If II "a^,^- I ^ matrix of the type n—l x it^ of which the determinants 

 are not all zero, and if mjc represent the value acquired by dk, when we at- 

 tribute to the indeterminates of that form the values of the determinants 



II ^^^^ II 



hi,A\ 



2=1, 2, . y=l, 2, . . w, 

 taken in the same order in which the determinants of any k horizontal rows 



of the matrix taken in forming the matrix of the numbers 



II J |i 



m^y m^, . . . are said to be simultaneously represented by the forms 

 Ky • • • K-v 



Let . . . Sj.^^r be a sequence of i' uneven concomitants, jUj^p i"t+2> 

 . • . being 0, or 1, but fj- and j^ij^i'^i being greater than 1 ; the uneven 

 numbers simultaneously represented by 9;+2> • • • ^i+i' ^^'^ ^^^^^ 

 to render the unit 



s=i+i' s=i+i' *^ s=i+i' g2_i 



( — 1) X(-l)*=*+i X (- 1)^=^+1 ' 8 



(which we shall symbolize by xp (z, i')), equal to + 1, or else are all such as 

 to render that unit equal to — 1. We therefore attribute to the sequence 

 of forms 9,.,.i . . . the simultaneous character \p (hi')= + l, or 



\p (i, i') = — 1, according as the former or latter of those equations is satis- 

 fied. If i' = \, the sequence consists of but one form, so that the charac- 

 ter xp (i, i') ceases to be a simultaneous character; in fact, if /.{._j,^=l, it 

 coincides with the supplementary character attributable to 0.^^ by III. ; if 

 yL(^.^j = 0, it either becomes nugatory {i. e. identically equal to +1, irre- 

 spective of the value of ^Wj+i), or it coincides with the supplementary cha- 

 racter of ^j.^j, according as that form (by IV.) has not or has a supplemen- 

 tary character. 



The complex of all the particular characters (principal, supplementary, 

 and simultaneous) constitutes the complete character of the system of con- 

 comitants . . . ^n-v ^very complete generic character, assignable 

 d priori, corresponds to actually existing forms, but only such characters 

 as satisfy a certain condition of possibility. This condition is expressed 

 by the equation 



;f.(0, ^-i)x n M)= + i, (A) 



in which, if is an even form, we understand by the symbol tlie 



quadratic character with respect to I^ of the half of any number, prime to 

 Is, which is represented by 0^. The unit \p (0, n—i) is formed in the same 



