354 report — 1865. 



of which the left-hand member may be written in another form. Instead of 

 counting the roots which appertain to the same value of A, and then summing 

 with respect to A, we may count the roots which appertain to the same value 

 'of a, and then sum with respect to c. If F(N) is the number of uneven 

 classes (primitive or derived) of determinant — N (we again count \ for a 

 class of det. — 1, or for a class derived from such a class), 2F(n — cr 2 ) will be 

 the number of roots appertaining to either of the values + c or — <r. We 

 thus obtain, finally, 



2F(«)+4F(n-l=)4-4F(ji-2 2 ) + 4F(n-3 2 ) + . . . =$(»)+¥(«), 



which is the formula V. 



132. "We shall also demonstrate (but with less detail) the formula VII. 



Writing x for u, and - for v, in the function/ (n, «, v), where n ^ — 1, mod 8, 

 x 



and multiplying by #*<"), we obtain an integral function of order 2#(«), 



which we shall designate by/(.r)- This function is not divisible by x, for 



f(0) = 1 ; but we shall now show that it is divisible by 



and that the quotient is prime to x"— 1. For this purpose we shall first 

 determine the index X, for which lim / ^' ; N , .r= 1 is finite and different 



LO-i) A J 



from zero. Let x=tp(-Y so that 



/(.r) = n 



, yl+16*' 



7 

 and let the positive quantity <r increase without limit ; then, ultimately, 



0(y=*K>V)=l-^O>), and l-.r=^(,V). 



Also, if $ is the greatest common divisor of 161- and y , and if a, b, I' are 

 determined by the equations 



a= — l^t b="L S'— - 



a ' a' ° r 



while c and d are two numbers (of which d is divisible by 2) satisfying the 

 equation ad—bc=l, we find 



yi + 161- 



_dy-\-citr 



y-L+16* S ' 



(t + b -, 



y- + 16fc 



whence solving for -?—, , and applying the formula ii. of Table A, 



