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