1867.] Prof. H. J. S. Smith on Quadratic Forms ^c. 



205 



but in this case v is even, and the sum of the series -L 2 is known. 



(ii) The factor n 1/ requires no explanation ; it is rational 



when n is mieven, and is a multiple of V ^ when n is even. 



(iii) The factor B„ is determined by the equations 



where (j^, i^3» • • • are again the fractions of BernouUi, so that /)i=^, 

 /33 = Jo-, etc. ... . . 



(iv) The factors (i) and (ii) depend only on the invariants and on the 

 number of the indeterminates, the factor (iii) only on the number of iu- 

 determinates. These factors are therefore the same for all genera of the 

 invariants 1^ I^, . . In-i. But the two remaining factors involve, or may 

 involve, certain of the generic characters, and are therefore not always the 

 same for all genera. In the factor n . x(^) the sign of multiplication extends 

 to every uneven prime I, dividing any one or more of the invariants 1^ I^, 

 .... I„_i : it will suffice therefore to define the function x(f^}> which de- 

 pends on only one of those primes. Let z\, i.,, ... be the indices of all 

 the invariants which are divisible by I ; let these indices be arranged in 

 order of magnitude, beginning with and ending with n (because lo and 

 In may be considered as divisible by ^). The positive differences 

 is+i—is we shall term intervals. By the moiety of any whole number a 

 we understand 4 a when a is even, | {a — I) when a is uneven. Let ks be 

 the moiety of the interval is+i — is ; when that interval is even, let the 

 barred symbol ks represent the product (— l)'''*In-;^ X Ig^^-^ x . . . x 



and let va-.) = 1 + ^' ^ ^'^^ ^is+i ^}^ Lastly, let a(/0 represent the 



product II (^1— i let a be the moiety of n — l,and fi the number of the 

 invariants Ij, I,,. . . which are divisible by 3. Then x(c^) is the integral 

 function of ^, defined by the equation 



when n is uneven, and by the equation 



when n is even. If D is divisible by ^, the symbol ^5 j is zero. In both 



formulae the sign of multiplication II extends to every value of us or the 

 value +1 is, as before, to be attributed to the symbols 0^ and Q^. 



VOL. XVI. U 



