ON THE TEIPLE THETA-FUNCTIONS. 
[From the Messenger of Mathematics, vol. vu. (1878), pp. 48—50.] 
As a specimen of mathematical notation, viz. of the notation which appears to 
me the easiest to read and also to print, I give the definition and demonstration of 
the fundamental properties of the triple theta-functions. 
Definition. 
A ( U, V, W) = 2 exp. ®, 
where 
® = (A, B, G, F, G, H) (l, m, ny + 2(U, V, W)(l, m, n), 
2 denoting the sum in regard to all positive and negative integer values from 
— oo to -foo (zero included) of l, m, n respectively. 
A ( U, V, W) is considered as a function of the arguments ( U, V, W), and it 
depends also on the parameters (A, B, G, F, G, H). 
First Property. A ( U, V, W) = 0, for 
U = \ {xiri A (A, H, G) (a, /3, 7)}, 
V = ±\y*i + {H, B, F)(a, /3, 7)}, 
W= \ {ziri + (G, F, G) (a, /3, 7)}, 
x, y, z, a, A, 7 being any positive or negative integer numbers, such that ax A Ay A 7z 
= odd number. 
Demonstration. It is only necessary to show that to each term of A there corre 
sponds a second term, such that the indices of the two exponentials differ by an odd 
multiple of 7ri.