753] ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 
243 
with the new characters ¡x + p! and v + v is, save as to an exponential factor, equal 
to the function © (u + 2-sr / ; p, v) with the original characters ¡i, v, but with the new 
arguments u + %a'. 
Notation. 
This is in some measure a development of the notation employed in my “ Memoir 
on the Theory of Matrices,” Phil. Trans, t. cxlyiii. (1858), pp. 17—37, [152] I use 
certain single letters u, etc. to denote sets or rows each of p letters, u = {u 1} ..., u p ): 
or if, to fix the ideas p — 3, then u = (u ly u 2 , u 3 ), and so in other cases. 
But I use certain other letters a, etc. to denote squares or matrices each of p 2 
letters ; thus, if p = 3 as before, 
a = 
in parentheses 
(«) = 
tt n , 
«12, 
«13 
y 
a 21 , 
«22, 
«23 
$31) 
«32, 
«33 
matrix is 
denoted by the same 
tin, 
«21, 
«31 
®12, 
«22, 
«32 
«13, 
«23, 
«33 
u, 
= («1, 
u 2 , 
u 3 ) and v, =■ (v l , v 2 
row (w, + v x , u 2 + v2, «3 + V3): and in like manner the sum a + b of the two matrices, 
or square-letters a and b, denotes the matrix 
«n + 2*ii, 
«12 T ^12, 
«13 + b 13 
«21 + ^21, 
«22 + b 22 , 
«23 1“ b 23 
«31 + b-.a, 
«32 + ^32, 
«33 + b 33 
and similarly for a sum of three or more terms. 
The product uv, =(u 1} u 2 , u 3 )(v u v 2 , v 3 ), of the two row-letters u, v denotes the 
single term u 1 v 1 + u 2 v 2 + u 3 v 3 . We have uv — vu. 
The product 
au, = 
«111 
«12, 
«13 
«21, 
«22, 
«23 
«31, 
«32, 
«33 
(u u u 2 , u 3 ), 
of a preceding square-letter a and a succeeding row-letter u, denotes the set or row 
(a n , a 12 , a 13 )(«i, u 2 , u 3 ), (a 21 , a^, a.P){;u u u 2 , u 3 \ (a 31 , a 32 , a^){u lf u 2 , u 3 ); 
the notation ua is not employed. 
31—2