663] 
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 
173 
We have, after Göpel (l.c. p. 283), a table showing how the ratios of the double 
^-functions are altered, when the arguments are increased by the quarter-periods 
A, B, A + B, K, L, K + L, 
that is, when u, u are simultaneously changed into u + A, u + Ä or into u + B, u +- B' 
etc. If instead, we consider the squared functions, the table is very much simplified, 
inasmuch as in place of the coefficients ± 1, ± i, it will contain only the coefficients 
+ 1 : and we may complete the table by extending it to all the combinations 0, A, B, 
A + B, K, K + A, K + B, K + A + B, L, L 4- A, L + B, L ■+• A + B, K + L, K + L + A, 
K + L + B, K + L + A + B of the quarter-periods : we have thus a table included in 
the annexed Table III., viz. attending herein only to the capital letters P, Q, B, S, the 
sixteen columns of the table show how the ratios of the terms — S 2 2 , — Si, etc., of the 
first column are altered when the arguments are increased by the foregoing combinations 
of quarter-periods, as indicated by the headings 0, A, B, etc., of the several columns. 
THE SQUARED DOUBLE ^-FUNCTIONS. 
1 
L + A 
L + 
B 
L + A 
4-B 
K+L 
K+ L 
+ A 
K + L 
+ B 
K+L+A+B 
-Qi 
-f 
-Qi = - 
cd 
- Q 1 = 
e 
Qi- 
— ab 
Pi- 
bc 
Pi- 
bd 
P 2 - 
ad 
Pi- 
ac 
Qi 
= ab 
Q 2 =- 
e 
Qz 2 = 
cd 
Qi- 
-f 
Pi- 
ac 
P 2 = 
ad 
Pi- 
bd 
Pi- 
bc 
Pi 
— ac 
P i = 
ad 
Pz 2 - 
bd 
Pi- 
bc 
II 
C? 
ab 
Q 2 = 
— e 
Qi- 
cd 
Qi - 
-f 
F i 
= ad 
Pi 2 = 
ac 
Pi= 
be 
P 2 _ 
1 3 — 
bd 
-Q 2 - 
e 
-Qi- 
— ab 
-Qi- 
f 
-Qi- 
— cd 
s 2 
= ae 
Si 2 =- 
b 
Si = 
— a 
<S' 3 2 = 
be 
-IP - 
d 
- Ri = 
c 
-Ri- 
— de 
— Ri -- 
— ce 
- s 2 
- a 
-Si=- 
be 
~S' 2 = 
- ae 
-£i 2 = 
b 
Ri = 
de 
Ri- 
ce 
R 2 = 
-d 
Ri- 
— c 
-Si 1 
-ft 
-8' 2 =- 
ae 
¿3“ = 
— be 
-S 2 2 = 
a, 
- Ri = 
c 
— R 2 = 
d 
- Ri - 
— ce 
-Ri- 
— de 
- Pi 
= C 
-K = 
d 
-Rz 2 = 
- ce 
-Ri = 
- de 
- Si = 
b 
-S 2 = 
— ae 
-Si = 
- be 
-Si = 
a 
- B? 
= d 
-Ri= 
c 
- R 2 = 
- de 
-Ri- 
— ce 
S 2 = 
ae 
Si = 
b 
Si = 
— a 
Si = 
be 
-Q 2 
= e 
-Qi = - 
ab 
-Qi = 
f 
-Qi- 
— cd 
P 3 = 
ad 
Pi- 
ac 
Pi- 
bc 
Pi- 
bd 
Ri 
II 
s. 
ct> 
R 3 2 = 
ce 
№ = 
-d 
Ri- 
— c 
- £f = 
a 
-- Si = 
-be 
— S 2 - 
— ae 
-Si = 
b 
Ri 
= ce 
Ri= 
de 
Ri= 
— c 
R 1 = 
-d 
Si- 
be 
S 2 = 
— a 
Si = 
-b 
S 2 = 
ae 
Qi 
= cd 
Qi = - 
f 
Qi = 
ab 
Q 2 = 
— e 
Pi- 
bd 
Pi- 
bc 
Pi- 
ac 
P 2 - 
ad 
Ss 2 
— be 
Si=- 
a 
S 2 = 
-b 
S 2 = 
ae 
Ri- 
ce 
il 
ce’ 
de 
Ri- 
— c 
R 2 - 
-d 
Pz 2 
— bd 
Pi = 
be 
Pi- 
ac 
p 2 = 
ad 
Qi- 
cd 
Qi- 
-f 
Qi- 
ab 
Q 2 - 
— e 
Pi 
= be 
Pz 2 - 
bd 
p 2 = 
ad 
Pi- 
ac 
- Qi- 
f 
— Qi — 
- cd 
-Q 2 = 
e 
-Qi- 
- ah 
a f 
be 
ae 
bf 
de 
ce 
df 
çf