206 Prof. H. J. S. Smith on Quadratic Forms ^c, [Dec. 5^ 



(v) Each factor x(^) of the product II . x(^) thus depends on an un- 

 even prime d dividing the invariants, on the indices of the invariants divi- 

 sible by on the principal generic characters with respect to d, and on 

 the quadratic characters with respect to 8 of the invariants not divisible 

 by S. The remaining factor niay be said to depend on the relation of 

 the concomitants and invariants to the prime 2 and its powers. The de- 

 termination of this factor presents no theoretical difficulty ; but on account 

 of the multiplicity of the cases to be considered, we shall confine ourselves 

 in this place to the two cases in which the invariants are all uneven. 



(A) When the invariants are all uneven, and the given genus is of an 

 uneven order, let 2,, represent the unit ( — 1)'* -^1^(0, n—1), where \p(0, n—1) 

 is the simultaneous character of the given genus, and h is determined by 

 the equation 



4h=(l- i)(I, + 1) + (I ~ 1)(U + + (IJ3- 1)(I,I,4- 1) 



-[-(y,-i)(ij3i,+i)+ .... 



+ ( In-i 1^-2- 1)( .... ln~3 !«-!+ 1). 



The value of i^n then is 



(1) if n=4A, 



l_[22A-i + (_l)A2j,or I, 



22\ 



according as D=l, or =—1, mod 4 

 (2) ifw=4\ + 2. 



A[22A+(_I)ASJ, orl. 



according as D=l, or =—1, mod 4 ; 



(3) ifn=:4A-f-l, 



i,[22-+(-i)^: 



(4) ifn=4\ + 3, 



D-1 



22A + ] 



(B) When the invariants are all uneven, and the given genus of an even 

 order, so that n=2v is even, the value of i^n^s 



It is easy to apply these general formulae to particular examples ; but 

 our imperfect knowledge of quadratic forms containing many indetermi- 

 nates, renders it practically impossible to test the results by any independent 

 process. The demonstrations are simple in principle, but require attention 

 to a great number of details with respect to which it is very easy to fall 



