AND DOUBLE THETA-FUNCTIONS. 
947 
© 00 
00 ~ 
10 
01 
11 
10 01 11 (2m) = 
X Y Z W i • 00 10 - 
-- that is © 00 (2w)=X, © rin = Y, &c. 
00 
« l 1 '"(2 !t )=X,. . 
® *0 *1 
00 
10 
01 
11 
fo fl (2 u) = 
E F G H ,i , • f 0, . ^ „ 
- that is ®q ( - ) (2m) = E, &c. 
© 01 
00 
10 
01 
11 
1* 
of If (2 u) 
I J K L The functions of (2 u') are denoted in like 
qT manner by accented letters 
1-1 00 , ,, , r/ 0 
2 8 00 (2«)=X,&C. 
© H 
2 2 
00 
3 1 
22 
M ff ( 2 m) 
10 
01 
11 
= M X P _Q 
0 
1 
2 
3 ! 
85. To simplify the expression of the results, instead of in each case writing down 
the suffixes, I have indicated them by means of the column headed “ Stiff.” 
Thus 
| 8-0 
S^u+v/. A-m'=XX' +YY' + ZZ' +WW' 
Suff. 
I 2 I 
means that the equation is to be read 
=X 3 X 3 '+YoY/+ZoZ;+Wo W/. 
It is hardly necessary to mention that the [ 8 — 0 | of the left hand column shows 
the current numbers of the theta-functions ; viz.: the left hand side of the equation is 
& 8 (m+m') . S- 0 (u — u'). 
And by a preceding remark the single arguments u-\-u' and u—u' are written in 
place of (m -j- u , v-\-v) and (u — u, v — v) respectively. 
6 f 2 
