172 FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. [663 
Referring now to Gopel’s memoir, Crelle, t. xxxv. (1847), pp. 277—312, we have 
the sixteen double ^-functions 
jP, jPj, Po, P 3 ; iQ, Qi, ïQo, Q 3 \ iR, iR\, R>, R 3 ; 
S, iS u iS 2 , S 3 , 
where the six functions affected with the i (= V — 1) are odd functions, vanishing for 
the values u = 0, u = 0 of the arguments. It is convenient to take x, oo as the 
values of x, x corresponding to these values u — 0, u — 0 : the expressions {«} will 
thus correspond to the six squares — Q 2 , — Qi, — R 2 , — Ri, — Si, — S 2 2 , and the ex 
pressions {ab} to the remaining ten squares P 2 , Pi, ..., Si; and after some tâtonnement, 
I succeed in establishing the correspondence as follows 
Si, Si, Ri, R 2 , Q 2 , Qi, Qi, 
= {«}> [b], {c}, {d}, {6}, {/}, {ab}, 
Pi, P 2 , S 2 , P 2 , P 2 , S 3 2 , Qi, Ri, Ri, 
{ae}, [ad], {ae}, {be}, {bd}, {be}, {cd}, {ce}, {de}, 
viz. the sixteen squared double ^-functions are proportional to the sixteen expressions 
— {a}, + {a&}, as hereby appearing. 
Table III. of the sixteen forms of 
0 
A 
B 
A+B 
K 
K 4- 
l 
K+B 
K + A+B 
Si 
- Si = a 
-Si = 
— be 
-S 2 - 
— ae 
— Si — 
b 
+ Ri = de 
Pi- 
ce 
R 2 = - 
d 
Ri- 
— c 
Si 
-Si ^b 
— S 2 = 
- ae 
-Si = 
- be 
- Si = 
a 
- Ri = c 
-R 2 = 
d 
— R 3 2 ----- — 
ce 
— Ri = 
— de 
Ri 
- Ri^c 
-R 2 = 
■+• d 
-Ri- 
- ce 
-Ri — 
de 
1 
Or„ 
il 
Cr- 
-S 2 =- 
- ae 
-s 3 2 =- 
be 
a 
R 2 
- R 2 =d 
-Ri= 
+ c 
- Ri - 
— de 
- Pi-- 
ce 
S 2 — ae 
¿V 2 = 
-b 
— 
a 
Si = 
be 
Q 2 
- Q 2 = e 
— = 
— ab 
— Qi — 
+/ 
-Qi — 
cd 
P 2 = ad 
Pi- 
ac 
Pi- 
bc 
Pi- 
bd 
Qi 
- Qi = f 
-Qi = 
— cd 
-Q 2 = 
+ e 
— Qi — — 
ab 
Pi = bc 
Pi- 
bd 
P 2 - 
ad 
Pi- 
ac 
Qi 
Qi = ab 
Q 2 = 
— e 
Qi- 
cd 
Qi — 
f 
Pi - ac 
p 2 = 
ad 
Pi- 
bd 
Pi- 
bc 
Pi 
Pi = ac 
P 2 = 
ad 
Pi- 
bd 
Pi = 
bc 
Qi = ab 
Q 2 = 
- e 
Qi- 
cd 
Qi- 
~f 
p 2 
P 2 = ad 
Pi - 
ac 
Pi = 
bc 
Pi- 
bd 
- Q 2 —e 
-Qi- 
- ab 
— Qi — 
f 
- Qi = 
— cd 
S 2 
S 2 = ae 
Si = 
-b 
Si- 
- a 
Si = 
be 
"8 
II 
1 
- Ri- 
c 
— Ri = - 
de 
-Ri- 
— ce 
pi 
1 2 
Pi = be 
Pi- 
bd 
P 2 = 
ad 
Pi- 
ac 
- Qi-f 
-Qi- 
- cd 
-Q 2 - 
e 
-Qi- 
— ab 
Pi 
P. 2 = bd 
P2_ 
J 2 — 
be 
Pi = 
ac 
P 2 = 
ad 
Qi — cd 
Qi- 
~f 
Qi- 
ab 
Q 2 = 
— e 
Si 
Cc 
05 ts 
II 
Si = 
— a 
Si- 
-b 
S 2 = 
ae 
O 
1! 
^ ce 
Ri = 
de 
Ri- 
c 
R 2 = 
-d 
Qi 
Qi = cd 
Qi- 
Qi- 
ab 
Q 2 — 
e 
"S 
-<5 
II 
Pi- 
bc 
Pi- 
ac 
P 1 = 
ad 
Ri 
II 
« 
Ri = 
de 
R, = 
— c 
R 2 = - 
-d 
II 
Si = 
- a 
Si=- 
b 
S 2 = 
ae 
Ri 
Ri = de 
Ri- 
ce 
R 2 = 
-d 
Ri — 
c 
- Si = a 
Cf 2 
o 3 — 
- be 
-S 2 =- 
ae 
—Si = 
-b 
OO 00 
cd 
e J 
ab 
bc 
bd 
ad 
ac