[717 
717] ON THE TRIPLE THETA-FUNCTIONS. 49 
belonging to 
Second Property. If U 1 , V 1 , W 1 denote 
lis system of 
are 
U + xiri + (A, H, G)(a, /3, 7), 
V + 2/7n + (.H, B, F) (a, /3, 7)» 
W+ziri + (G, F, C)(a, ¡3, 7), 
respectively, where x, y, z, a, /3, 7 are any positive or negative integers (zero values 
admissible), then 
A n + 7); 
Vi> W 1 ) = ex V .{-(A,B,C,F,G, H)(a, /3, 7 ) 2 }. exp. {- 2 (aU+ f3V+ ryW)}.*(U, V, W), 
= exp. {— (A, ...) (a, /3, 7 ) 2 } . exp. {- 2 (otU + /37+ 7 IF)}. *(*7, V, W). 
once written 
Demonstration. Writing ^r(U u V 1 , W x ) = 2. exp. (yfi, then in the expression of © : 
we may in place of l, to, n write l — a, to — /3, n — 7; we thus obtain 
@i = 01, to — ft, n — 7) 2 + {( l - a)[U + anri + (A,...)(«, /3, 7)] 
+ (to - /3) [7 + yiri + {H, ...) (a, /3, 7)] 
+ (w -7) [17 + 27™ + (6r, ...)(a, /3, 7)]}, 
which is 
= (A, ...)(Z, to, ?i) 2 
+ 2 (¿ZT’+TOF+ftTF) + 2 (Ix + my + nz) iri + 2(H, ...) (1, to, ft)(a, /3, 7) 
-2 04, ...)(Z, to, w)(a, /3, 7) 
— 2 (a7+/37 +7IF) —2 (a#+ /32/4-72)77-1 — 2(^4, ...)(a, /3, 7) 2 
+ (-4, ...)(«> /3, 7) 2 , 
which is 
= (A, to, ft) 2 + 2(IU+ to7+ ftIF) 
-(A ...)(«, A 7> 2 — 2(a£7 + /37+ 7IF) 
+ 2 [(£ - a) a; + (to — /3) y + (ft — 7) s] iri. 
Hence, rejecting the last line, which (as an even multiple of 7ri) leaves the exponential 
unaltered, we see that ^s{Ux, 7 a , W x ) is =^-(£7, 7, IF) multiplied by the factor 
exp. {-(H, ...) (a, /3, 7)2}. exp. {-2 (a 7+/37+7 IF)}, 
which is the theorem in question. 
In many cases a formula, which belongs to an indefinite number s of letters, is 
most easily intelligible when written out for three letters, but it is sometimes con 
venient to speak of the s letters l, or even the s letters l,..., n, and to write 
out the formulae accordingly. 
r clearer, as 
lemonstration 
c. xi. 7