ON THE THEORY OP NUMBERS. 339 



is, by a known theorem, — or — — , according as 0-<*<7r, or ir-<a;<27r. 



Hence attributing to n only uneven values, and denoting by »i' and m" the 

 values of w» inferior and superior to 2a, 



V=: 



/a^LW v«")J 



8\/a 



4v'a \w7' 



because (— 7)=— (^^ iV 



\w»7 \2A + m'/ 



If we attribute to n all integral values, the equation 



sin X , sin 2x , sin 3ar 



1 / X sin ar , sin 2x , sin 3x , 



-(TT — x)= + \- H 



2^ ^ 12 3 



which subsists for all positive values of x less than 2ir, will give the value 

 already obtained for V by the former method, viz., 



2\/A \m/ 



The mode of application of this method may be still further varied ; for, 

 instead of substituting for (—1) ^ | — 1, we may leave the factor (—1) " 



unchanged, and substitute for (— ), by means of the equation 



which, as well as the substitution which we have employed, is deducible from 

 the formulae of Gauss*. We should thus obtain a third expression for V, 

 different in form from both of those which we have already found. 



The forms which the expression of h can assume are very numerous; we 

 select the following as examples, D still denoting a determinant not divisible 

 by any square. 



I. IfD^l, mod 4. 



For a positive determinant, D=A, 



k=(l) ^ ^_ sffl log tan ("^\ 



VD/iog(T+UVD) \V ^ \2DJ 



For a negative determinant D = — A, 



the summations extending to every uneven value of m prime to A and less 

 than A. 



II. If D be not = 1, mod 4. 

 For a positive determinant, 



*" - log(T+UVD) ^ Q ^°S «- (^) 



* See equation (2) of the preceding note. 



z2 



