48 
ON A THEOREM OF M. LEJEUNE DIRICHLET S. 
[Ill 
then [-) = + or — according as n is or is not a quadratic residue of p (or, what is 
the same thing, p being positive, (modjp)), and for P=pp'p" 
and the summation extends to all the values of n, n' of the form above mentioned. 
In the particular case D — —1, it is necessary that the second side should be doubled. 
The method of reducing the equation is indicated in the memoir. The following are 
a few particular cases. 
D = —1, 
or 
^q x -+y~ = 4-N (_)!(«—i) qnri, 
(1 + 2 q i + 2q 16 + 2 q 36 + ...) (q + q 9 + q 25 4- q® + ...) 
2 s l 
<? 
+ ... 
D = - 2, 
or 
1 — q 3 1 — cf ' 1 — q 10 1 — q 1 
Sqrf+w 2 — 2X (—^i(w-i)+i(n 2 -D q-nri, 
(1 + 2q 2 + 2q 8 + 2q 18 ...)(q + q 9 + q r ° + q i9 + ...) 
<1 
+ 
q 7 
1 — q% \ — q$ l — gi 
an example given in the memoir. 
1 - q u 
+ &c. 
D = — 3, 
V ,+3J/2 =2S^ q nn ’, 
(q 1 + q 25 + q i9 + q 121 + q lw ...) (1 + 2q 12 + 2q ls + 2q m ...) 
+ 2 (q 3 + (f + q 75 + q u7 + ...)(q 4 + q 16 + q M + q m ...) 
qii + q55 
q + q* q 5 + q 25 q 7 + q 3 
F-q 6 ~ F-q 30 + 1 -q* 
1 — q 6 
+ ... 
I am not aware that the above theorem is quoted or referred to in any sub 
sequent memoir on Elliptic Functions, or on the class of series to which it relates: 
and the theorem is so distinct in its origin and form from all other theorems relating 
to the same class of series, and, independently of the researches in which it originates, 
so remarkable as a result, that I have thought it desirable to give a detached state 
ment of it in this paper.