348 report— 1865. 



VI. 2F(m)-4F(m-l 2 ) + 4F(m-2 2 )-4F(m-3 2 )4-. . . 

 =(_l)K»»-i)[$(m)— ¥(m)]. 



VII. 2F(H0-4F(m-4 2 ) + 4F(m-8 2 )-4F())i-12 2 )-h... 

 = ( - l)iC>»-7)[*'(»»)— *'( m )]« 



VIII. 4S,< - 1 r ( '"- s2) [2F (^) -3Q(^) ] 



= (_l)^ m_l) r*'(m) - ¥'("») 1 • 



In these formula; mis any positive uneven number ; in the 1st, /* is >2 ; in 

 the 7th, m is = — 1, mod 8 ; in the 8th, m is =t}-1, mod 8, and the sunima- 



tion extends to all values of s for which 1 is integral and not negative ; 



similarly, the series, in the first seven formula; are to be continued until the 

 numbers affected with the signs F and G becomenegative. If n is any posi- 

 tive number, even or uneven, <!>(«) is the sum i of the divisors of n, ^(n) the 

 excess of those divisors of n which surpass V ' n above those divisors which 



are surpassed by >Jn; <J>'( Hi ) is the sum 2(^ W extended to all the divisors 

 of m ; ¥'(m) the excess of the sum 2( - \l extended to all the divisors of m 



which surpass Vm, above the sum s(^ W extended to all divisors of m 



which are surpassed by Vu>. Lastly, F(n) is the number of uneven classes, 

 G(n) the whole number of classes, of forms of determinant — n ; the classes 

 (1, 0, 1), (1, 1, 1), and their derived classes, being counted as \ and \ re- 

 spectively; to F(0) we attribute the value 0, to G(0) the value — T V*- 



The arithmetical functions F(») and G(«) satisfy the cquationsF(4n)=2F(«); 

 G(4»)=F(4h) + «(»); G(»)=F(«), if «=l,or2, mod 4 ; G(n)=2F(n), if 

 n=7, mod 8 ; G(»)=fF(»), if n=3, mod 8. With the help of these rela- 

 tions (which may be demonstrated by elementary considerations [see art. 113 

 of tins Report], but which may also be inferred from the theory of elliptic 

 functions) the formula; I. -VIII. may be transformed and combined in various 

 ways, so as to afford new and interesting results. Of these derived formula? 

 M. Kronecker has given two, 



IX. F(n) + F(n-1 . 2) + F(«-2 . 3) + F(n-3 . 4)+ . . 



=1*(4h+1), 



X. E(n) + 2E(ii-l 2 ) + 2E(»-2 2 ) + 2E()i-3 2 )+ . . 



=£[2+(-l)«]X(»), 

 where n represents any positive integer, X(«) the sum of its uneven divisors, 

 and E(»=2F(?*)— G(»), so that E(«) is a function satisfying the equations 

 E(4n) = E(»); E(n)=0, if »=7, mod 8; E(ji)=fF(n), if n=3, mod 8. 

 The first of these formula; is obtained by subtracting VI. from V. ; for 

 F(»— le Jc+l)=±T(4n + 1 — [21- + 1] 2 ). The other, if n is uneven, coincides 



* The right-hand members, of the formulae I.-VIII. are rendered simpler by this con- 

 ventional estimation of a class of det. — 1 as f , and of an improperly primitive class of det. — 3 

 as ^. We have already seen that this convention is a natural one (art. 126, note) ; it is, 

 however, less easy to interpret the assumption G(0)= — ^ g . M. Kronecker has given his 

 formulae in their complete expression when these conventional estimations are disregarded ; 

 in his subsequent notes, however, he seems to prefer the simpler form, which we have 

 adopted in the text. 



