342 report — 1865. 



the application of Dirichlet's method shows that, for every representation of 

 P hy <p, ( — l) a+/3 has the same value as ( — 1)* in either primitive represen- 

 tation of P by/, and, conversely, that for every representation of P by /, (— 1)* 

 has the same value as (— l) a+ ' 3 in either primitive representation of P by <j> ; 

 -whence the units ( — 1)* and (— l) a+/3 have all the same value, and 2(— 1) A 

 = 2(— l) a+p = +(V + 1). The ambiguous sign is that of ( — 1)* in the pri- 

 mitive representation of P by/, and will be found (by reasoning similar to that 



which establishes Gauss's first criterion) to coincide with ( — 1) e", where 



p-i 

 e is the unit satisfying the congruence 2 4 =e, modjj. (ii)If P=p 2 ",£>=3, 

 mod8, there is but one representation of P by <p, and 2(— l) a ^=( — 1)" : 

 there are 2v + l representations of P by/, of which two are primitive, 2(v— 1) 

 are derived from the primitive representations of p 2 , p*, . . . p 2 0- J ), and in the 

 remaining one 6=0. Applying Dirichlet's method to the equation 



^=(4rt + l) 2 + 86 2 =(4a+l) 2 + 16/3 2 

 (in which <r=l, 2, . . v, (3=0, 4a + l = ( — l) "? 0- , and the representation by 

 /is primitive), we find (— 1)*=(— 1)°"; whence inasmuch as the character 

 ( — 1)* is the same in a derived representation, and in the representation from 



which it is derived 2(-l) 4 =l + 2Z(-iy=(-iy=-Z(-l) a+fi . (iii) Lastly, 



if 'P=pP v ,p=5, mod 8, there is but one representation by/, and 2( — 1)*= + 1; 

 there are 2»/+l representations by <p. Applying Dirichlet's method as in 

 the preceding case, Ave find that for any primitive representation of an even 

 power of p by <j>, (— l) a+/3 =-fl ; whence, for a derived representation in 

 which the greatest common divisor of the indeterminates is (f, (— l) a+ ' 3 



v-1 

 = (-lf. Consequently S(-l) a+ ' 3 =2 S (-l) a +(-l)"= + 1=2(-1)*. 





 This completes the demonstration of Jacobi's theorem. 



Let P be any uneven number and ^(P) the positive numerical transcendent 

 defined by the equation 



rf_l (P-l d-\ rfg-1 



X 2 (P)X^(P) = S(-1)~X2(-1)" 8 XS(-l) 2 8 , 

 where ^(P) is the number of divisors of P, and d is any divisor of P. It 

 will be seen that X (P)=0, except when P is capable of representation both 

 by © and /: when P is capable of such simultaneous representation, let 

 P = (4a + l) 2 + 16/3 2 be a representation of P by in which the greatest 

 common divisor of 4a + 1 and (3 is the least possible ; let or = 4a 4-1 + 4*73, and 



let [" +? I represent the quadratic character (art. 27) of l + i' with respect 

 ■m J 



to to- ; the equation 



S (-i/ =s( -ir^[^]x(f) 



will hold in each of the cases considered separately above ; but the nume- 

 rical functions occurring in this equation satisfy the condition (^(PJ X ^(P 2 ) 

 =0(P 1 P.,), where P i; P 2 are relatively prime ; the equation is therefore uni- 

 versally true for every uneven number P, and implies the identity 



2(— i)"(-/ M '" +1 i 2+ii ""=s(— l)"' + "o ,4, " +1):!+16 " , =2; — — x(P)? P - 



