33S REPORT — 1861. 



this expression the sum 2( — 1) ^ ( — ) is zero, because the terms corre- 

 sponding to m and 2A + m destroy one another ; so that 



Distinguishing the two cases D=A, and D= — A, and observing that the 

 imaginary parts vanish identically, as they ought to do, because V is real, we 

 have, finally, if D= A, 



and if D= — A, 



m— 1 



2Va ^ ^ \A/ 

 2v'A \m) 



\m. 



(ii.) The series S( — )- can also be summed by substituting for / — ) its 



trigonometrical value deducible from the formulae of Gauss. We will take 

 as an example the case in which D= — A^3, mod 4'. Writing n for m, and 

 m for w, in the equation 



^ e*A )=^x (-1) (^^jVA, 



we find, observing that ^(1 + a) is uneven, 



2Va \m) \ 4A / 

 the summation extending to every value of m prime to 4A and less than it. 



Substituting this expression for | — j in V, we have 



V=-l-SfP^2lsinf2mn.^ 

 '2VA \m) n \ 4A / 



Since the expression which we have substituted for | — | is zero, when n is 



not prime to ^A, we may attribute to n, in the series 



either all uneven values, or all integral values. The sum of the series 



sin a; .sin 3ar , sin 5x , 

 13 5 



