1864.] 



containing more than Three Indeterminates, 



203 



prime dividing ^lil-i — ^ ; we infer from the identity (B) that the 



Vi Vs-i 



numbers prime to which can be represented by 0^, are either all qua- 

 dratic residues of 3% or all non-quadratic residues of Zi. In the former 



case we attribute to / the particular character ^.^^=4-1 ; in the latter 



the particular character ^y^= — 1. If Vi=lj ^. if the form itself 

 do not admit of any primary divisor beside unity (which is the only 

 important case), the product / J^^-^^^X f'^^^^^X . . . . 



\Vn-l Vn-2/ \Vn-2 Vw-3/ 



is equal to - ; whence, inasmuch as every prime that divides Vn also 



Vn-l 



divides — it appears that a primitive quadratic form will always have 



one particular character, at least with respect to every uneven prime 

 dividing its determinant, and will have more than one if the uneven 



prime divide more than one of the quotients -f- 



Vi Vi-i 



The subdivision of an order into genera can now be effected by assign- 

 ing to the same genus all those forms whose particular characters co- 

 incide. But it remains to consider whether the above enumeration of par- 

 ticular characters is complete. It is evident that we might apply the 

 theorem (B) to other concomitants besides those included in the funda- 

 mental system ; and it might appear as if in this manner we could obtain 

 other particular characters besides those which we have given. But it can 

 be shown that such other particular characters are implicitly contained in 

 ours ; i, e. it can be shown that two quadratic forms, which coincide in 

 respect of the particular characters deducible from their fundamental con- 

 comitants, will also coincide in respect of the particular characters dedu- 

 cible from any other concomitant. Again, it will be found that if the 

 determinant be uneven, there are no particular characters with respect to 

 4 or 8. For this case, therefore, our enumeration is complete. But 

 when the determinant is even, besides the particular characters arising from 

 its uneven prime divisors, there may also be particular characters with 

 regard to 4 or 8. There is no difficulty in enumerating these particular 

 characters ; nevertheless we suppress the enumeration here, not only 

 because it would require a detailed distinction of cases, but also because 

 there appears to be some difficulty in showing that the characters with 

 regard to 4 or 8, which may arise from the excluded concomitants, are 

 virtually included in those which arise from the concomitants of the fun- 

 damental set. 



