208 Prof. H. J. S. Smith on Quadratic Forms ^c. 



[Dec. 5, 



(2) In every other case, 



(4«12^A) =Q,(4-11^A) X >; X 2(^^^<5~4D), 



where = or h, according as a = 0, or a^O. 

 (B) for seven squares. 



(1) If A=3, mod A, 



*,(4«i2'A) = Q,(4-0-)A X r/ X A)(25- A), 



where 7? = 30, if a = 0, A=3, mod 8 ; 7? = |x 37, if a=0, A=7, mod 8 

 7/=iXl40, if a>0. 



(2) In every other case, 



$'(4-i2-A) = Q,(4'^a-A X n X 2 ^^^s(5-2A)(*-4A), 

 where r}=\, or -f-^, according as a = 0, or a>0. 



4A 



The sums S, and 2 in these formula are easily reduced (by distinguish- 

 1 1 



ing different hnear forms of the number A) to others more readily cal- 

 culated (see the note of Eisenstein, to which we have already referred) ; 

 but in the present notice we have preferred to retain them in the form in 

 which they first present themselves. 



We shall conclude this paper by calling attention to a class of theorems 

 which have a certain resemblance to the important results established by 

 M. Kronecker for binary quadratic forms. 



Let 5 '^]^' represent the weight of the quaternary classes of the inva- 

 riants [1, 1, M] ; the weight of the senary classes of the invariants 

 [1, 1, 1, 1, M], then 



F,(M)+2F,(M-r) + 2F,(M~2^)+ ... =:2(-l)^^, 

 Fg(2M) + 2F,(2M-r) + 2F/2M-2^)+ . . . =^d\ 



In the first of these formulae JM is any unevenly even number, or any 

 number =3, mod 4 ; in the second M is anj uneven number : the series 

 in both are to be continued as long as the numbers M — or 2M— 5^ 

 are positive ; d is any uneven divisor of M. The origin of these formulae 

 (which may serve as examples of many others) is exactly analogous to that 

 which M. Kronecker has pointed out as characteristic of the more ele- 

 mentary of the two classes into which his formulse are naturally divided. 

 Whether, for forms of four and six indeterminates, similar formulae exist 

 comparable to the less elementary formulae of M. Kronecker. and whether, 

 for form.s containing more than six indeterminates, such formulae exist at 

 all, are questions well worthy of the attention of arithmeticians. 



