ON A THEOREM OF M. LEJEUNE-DIRICHLET’S. 
[From the Cambridge and Dublin Mathematical Journal, vol. ix. (1854), pp. 163—165.] 
The following formula, 
^qax 2 +2bxy+cy 2 ^qa'xi+ib'xy+c'y 2 e i(w 2 —l) 
.(3) 
is given in Lejeune-Dirichlet’s well-known memoir “ Recherches sur diverses applica 
tions &c.” (Crelle, t. xxi. [1840] p. 8). The notation is as follows:—On the left-hand 
side (a, b, c), (o', b', c'), ... are a system of properly primitive forms to the negative 
determinant D (i.e. a system of positive forms); x, y are positive or negative integers 
including zero, such that in the sum Xq axl+2bxy+c y 2 , ax 2 + 2bxy 4- cy 2 is prime to 2D, 
and similarly in the other sums; q is indeterminate and the summations extend to 
the values first mentioned, of x and y. On the right-hand side we have to consider 
the form of D, viz. we have D = PS 2 or else D = 2PS 2 , where S 2 is the greatest 
square factor in D and where P is odd: this obviously defines P, and the values 
of 8, e, which are always ± 1 (or, as I prefer to express it, are always +) are given 
as follows, viz. 
D = PS 2 , 
P = 1 (mod 4), 
s, 
e = + +, 
$2 
¡1 
P = 3 (mod 4), 
8, 
« = -+, 
D = 2 PS 2 , 
P = 1 (mod 4), 
8, 
€ = + -, 
D = 2PS 2 , 
P = 3 (mod 4), 
8, 
e = , 
n, n! are any positive numbers prime to 2D, \^pj is Legendre’s symbol as generalized 
by Jacobi, viz. in general if p be a positive or negative prime not a factor of n,