Paes 
is the sum of those numbers o, (m,n), belonging to the numbers 
r divisible by d. However these are evidently the terms of a fourth 
part of a system of rests according to modulus — multiplied by d. Now, 
the numbers k.dm giving after division by » the same rests as the 
numbers Am after division by = the last sum is equal to 
and so we get the equation 
n 
n (m,n) = Zu (d) A (m, ~), 
d d 
from which ensues reciprocally 
H (m,n) = = y(m, a). 
d 
‘ m 3 
If we define the generalised symbol ((=)) by the equation 
N(n)—1 
mn 
4 
= 5, (m, n) H (m, n) 
> ie 
i.e. by a corresponding generalisation of the lemma of Gauss, we 
have according to previous developments 
= 7 (m, d) 
d 
= (Ce) eae Weel 
hence this natural definition coincides with the exposition of JACOBI 
12 
Proceedings Royal Acad. Amsterdam. Vol. III. 
