1867.] Prof. H. J. S. Smith on Quadratic Forms ^c. 199 



is always to be regarded as preceded by an even invariant, and as fol- 

 lowed by an even invariant; similarly the forms 0^ and are to be re- 

 garded as respectively preceded and followed by uneven forms. Two even 

 forms cannot be consecutive in the series 9^ . . Q^-v 



The preceding observations enable us to assign all the orders which may 

 exist for any given invariants ; if the series of invariants I^, 1^ . . . \ 

 present w different sequences each consisting of an uneven number of un- 

 even invariants, preceded and followed by even invariants, there are 2"^ as- 

 signable orders. These orders, in general, all exist ; there are, however, 

 the following exceptions to this statement : — 



(1) If, the number of indeterminates being even and equal to 2y, D is 

 uneven, there is an assignable order in which the concomitants • • • ^n-i 

 are alternately even and uneven. But, as has been already said, this order 

 does not exist if D = — 1, mod 4 ; and, if the invariants are all squares, 

 it does not exist, even if D = 1, mod 4, unless the equation 



(-1) ' =(-1) ' 



(in which Jc is the index of inertia) is also satisfied. 



(2) If, the number of indeterminates being uneven and equal to 2i'+ I, 

 D is uneven, and even, there is again an assignable order in which the 

 concomitants 0^, . . . are alternately even and uneven. But, when I^^ 

 is the double of a square, and the other invariants are squares, this order does 

 not exist unless the equation (—1)^ *^"'~^'* = (— 1)^ ^''^""^-'(in which /c is still 

 the index of inertia) is satisfied. 



The reciprocal case (that obtained by changing and into ln-s> and 

 ^n-s> for every value of 5 from to n) presents a similar exception, which 

 it is not necessary to enunciate separately. 



The generic characters of the form 5^, or, more properly of the system of 

 concomitant forms 0^, 5^, . . . so far as they depend on uneven primes 

 dividing the invariants, have been already defined in the former notice, and 

 the definition need not be repeated here. These characters we shall term 

 principal generic characters of the system. When the invariants and 

 primitive concomitants are all uneven, the principal characters are the only 

 generic characters, with the exception of a certain character, which we 

 shall define hereafter, and of which the value is not independent of the 

 principal characters. In other cases, the forms of the concomitant system 

 may acquire generic characters with respect to 4 or 8 : these we shall term 

 supplementary. What supplementary characters exist in any given case 

 may always be ascertained by applying the following rules. In their enun- 

 ciation we represent by l'^ the greatest uneven divisor of 1; taken with the 

 same sign as I^, by . the exponent of the highest power of 2 contained in 

 Ij, increased by 1 if one of the two forms ^.^^ is even, and by 2 if 

 both those forms are even ; we suppose < z -< ??. 



