1807.] 



Prof. H. J. S. Smith on Quadratic Forms ^t. 



207 



into error. As soon as they can be put into a convenient form, they shall 

 be submitted to the Royal Society. 



Eiseustein has observed that, when the mmiber of in determinates does 

 not surpass eight, there is but one class of quadratic forms of the dis- 

 criminant 1, but that, when the mmiber of indeterminates surpasses eight, 

 there is always more than one such class. This observation is in ac- 

 cordance with our general formulae, except that they imply the existence of 

 an improperly primitive class of eight indeterminates and of the dis 

 criminant 1. 



The theorems which have been given by Jacobi, Eisenstein, and recentl} 

 in great profusion by M. Liouville, relating to the representation of numbers 

 by four squares and other simple quadratic forms, appear to be deducible 

 by a uniform method from the principles indicated in this paper. So 

 also are the theorems relating to the representation of numbers by six 

 and eight squares, which are implicitly contained in the development^ 

 given by Jacobi in the ' Fundamenta Nova.' As the series of theorems 

 relating to the representation of numbers by sums of squares ceases, for 

 the reason assigned by Eisenstein, when the number of squares surpasses 

 eight, it is of some importance to complete it. The only cases which have 

 not been fully considered are those of five and seven squares. The prin- 

 cipal theorems relating to the case of five squares have indeed been given 

 by Eisenstein (Crelle's Journal, vol. xxxv. p. 368) ; but he has considered 

 only those num.bers which are not divisible by any square. We shall here 

 complete his enunciation of those theorems, and shall add the corresponding 

 theorems for the case of seven squares. We attend only to primitive 

 representations. 



Let A represent a number not divisible by any square, 12'- an uneven 

 square, o any exponent. By ^.(4'^O-A), ^>-(4^~0-A), we denote the number 

 of representations of 4^0-A by five and seven squares respectively ; by 

 Q.(4'^0-A), Q-(4'^l^i''A), we represent the products 



5 X 2^^ X 12' X n 

 7 X 2''^ X X n 



the sign of multiplication n extending to every prime dividing U, but 

 not dividing A ; we then have the formulce 



(A) for five squares. 



(1 j If A==l, mod 4, 



^,(4«a-A) = Q-(4-0-A) X X sQ^.^O 



where, if A=l, mod 8, r] = 12 ; if A=5, mod 8, ?/ = 28 or 20, accordii 

 as (1 = 0, or f(>0. If, hov/ever, A = l, we are to replace r/x2 by 2. 



