AND DOUBLE THETA-FUNCTIONS. 
935 
anywise suggests the single-and-double-letter notation : this first presents itself in 
connexion with the relations between the functions given by the product-theorem : 
and as the product theorem is based upon the notation by characteristics, it is proper 
to present the theorem in this notation, or in the equivalent current-number notation: 
and then to show how by the relations thus obtained between the functions we are 
led to the single-and-double-letter notation. 
68. There are some other notations which may be referred to: and for showing the 
correspondence between them I annex the following table :—- 
The double theta-functions. 
Asterisk 
denotes the 
odd functions. 
1. 
Current 
number. 
2. 
Character. 
3. 
Single-and- 
double-letter, 
Cayley. 
4. 
GrdPEL. 
5. 
CrOPEl- 
Cayley. 
6. 
Rosen rain. 
7. 
W EIER- 
STRASS. 
8. 
Kummek. 
do 
q00 
"mO 
BD 
P'" 
P 8 
do 3 
d 5 
1 
12 
] 
10 
00 
CE 
R rl 
r 3 
32 
4 
8 
2 
01 
00 
CD 
Q'" 
Qs 
03 
01 
10 
3 
11 
00 
BE 
S'" 
S, 
33 
23 
6 
4 
00 
10 
AC 
P' 
Pi 
02 
34 
4 
* 
5 
10 
10 
C 
/R ' 
R, 
12 
3 
16 
6 
01 
10 
AB 
Q' 
Qi 
03 
0 
2 
* 
7 
11 
10 
B 
iS' 
Si 
13 
24 
14 
8 
00 
01 
BC 
P" 
p„ 
20 
12 
9 
9 
10 
01 
DE 
R" 
r 2 
30 
03 
5 
* 
10 
01 
01 
F 
»'Q" 
q 3 
21 
.02 
11 
11 
11 
01 
A 
»'S" 
s 2 
31 
13 
7 
12 
00 
11 
AD 
P 
p 
00 
0 
1 
13 
10 
11 
D 
i'R 
R 
10 
04 
13 
14 
01 
11 
E 
iQ 
Q 
01 
1 
o 
O 
_ 
15 
11 
11 
AE 
S 
s 
11 
14 
15 
