904 
PROFESSOR A. CAYLEY ON THE SINGLE 
where p, q have in the four products respectively the values (0, 0), (1, 0), (0, 1), and (1,1); 
© is written in place of 3- to denote that the parameters (a, h, b) are to be changed 
into ( 2 a, 2 h, 2b). It is to be noticed that if a, a' are both even or both odd then 
-|(a-|-«'), -|-(a— a') are integers; and so if /3, /3' are both even or both odd then 
-|(/3+/3'), -${/3—/3') are integers; and these conditions being satisfied (and in particular 
they are so if a = a, fi=/3') then the functions on the right hand side of the equation 
are theta-functions (with new parameters as already mentioned); but if the conditions 
are not satisfied, then the functions on the right hand side are only allied functions. 
In the applications of the theorem the functions on the right hand side are eliminated 
between the different equations, as will appear. 
13. The proof is immediate : in the first of the theta-functions the argument of the 
exponential is 
/m + « ,n + /3 \ 
\u + u'-\- 7, v + v' + SJ’ 
and in the second, writing m, n instead of m, n, the argument is 
fm' + a' ,n' + /3' \ 
\ u—io' + y', v — v' + S')’ 
hence in the product, the argument of the exponential is the sum of these two 
functions, 
=i(«, K ^X m + a > n +P ) 2 +2^’( m fi-a .u-\-u -\-y .-\-.n -\-/3 .v+ff+S) 
h, bXm a, n-\-n.u — u -\-y . -\-.n 
Comparing herewith the sum of the two functions 
/V + K a + a 0> ^ + i(^+/50\ /V + K a_ a 0> V ' + W3—P')\ 
\ 2 ^ 4-|-7 + 7 , , 2 r + 8 + 8 ' /’ \ 2 w' + 7 — 7 ' ,2v' + B — S' J’ 
— i( 2 a, 2 h, 2b'Jj^ J r\{ aJ r a ), 
+ i (2 a, 2 h, 2 bJp'+±(*—a), /T )) 3 
+ Tr7rt’{/r — ).2ll -\-y — y .-\-.v + ijr(/3— f3').2v +8 — S }, 
the two sums are identical if only 
m+m'= 2 /a, n-\-n—2v, 
m — m'=2/jL,', n—n'=2v, 
