900 
PROFESSOR, A. CAYLEY ON THE SINGLE 
Properties of the theta-functions : Various sub-headings. 
Even-integer alteration of characters. 
0. If x, y be integers, then m, n having the several even integer values from — oo 
to + 00 respectively, it is obvious that m + a + 2x, n-\-/3-\-2y will have the same 
series of values with m+a, n-\-/3 respectively; and it thence follows that 
3 
/« + 2x, (3 + 2y 
\7 > S 
Similarly if z, w are integers, then in the function 
3 
/« , /3 
\ 7 + 2 z , 8 + 2 w 
the argument of the exponential function contains the term 
\tti -hy+ 2z. + .n ; 
this differs from its original value by 
^vi(m-\-a..2z.-\-.n-\-/3.2iv ), 
= 7ri(mz-Vmv)-\-Tri(az-\-/3iv), 
and then, rn and n being even integers, 'mz-\-nw is also an even integer, and the term 
7 n(mz-\-nw) does not affect the value of the exponential: we thus introduce into each 
term of the series the factor exp. Tri(az-\-fiw), which is in fact =[—) aZ+PiC ; and we 
consequently have 
3 
,/3 
(u, v) 
y7 + 2s;, 8 + 2w} 
or, uniting the two results, 
/« + 2x, /3 + 2y\ . , 
*U+a., s + 2»j <“>») 
this sustains the before-mentioned conclusion that the only distinct functions are the 
16 functions obtained by giving to the characters a, (3, y, S the values 0 and 1 
respectively. 
Odd-integer alteration of characters. 
7. The effect is obviously to interchange the different functions, 
