663] FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 169 
Here the symbols be, ce, etc., denote differences ; [ce] is the product of four differences : 
the arrangement in two lines is for convenience only. 
We thus obtain the series of values of [a], [ab\ etc., which although only required 
as subsidiary to the determination of the corresponding values of [a}, {ab}, I nevertheless 
give in a table. 
The signs are given as they were actually obtained, but as we are concerned only 
with the ratios of the functions, it is allowable to change all the signs in any 
FOR THE SIXTEEN SPECIAL VALUES OF X, OC. 
be 
¥ 
cd 
ce 
¥ 
de 
df 
¥ 
+ 
ab 
ae 
+ 
ab 
af 
+ 
ac. 
ad 
+ 
ac 
ae 
+ 
ac 
df 
4* 
ad 
ae 
+ 
ad. af 
+ 
ae. af 
0 
0 
+ 
bc . 
bd 
+ 
bc 
be 
+ 
bc 
¥ 
+ 
bd 
be 
+ 
bd.bf 
+ 
be . bf 
- 
be . 
ce 
- 
bc . 
¥ 
0 
0 
0 
+ 
cd 
ce 
+ 
cd. df 
+ 
ce . cf 
- 
bd. 
de 
- 
bd. 
¥ 
0 
- 
cd 
de 
- 
cd. 
df 
0 
0 
+ 
de . df 
0 
- 
be . 
¥ 
+ 
ce . 
de 
0 
- 
ce . 
¥ 
0 
- 
de. ef 
0 
+ 
¥ ■ 
¥ 
0 
+ 
¥■ 
df 
+ 
¥■ 
¥ 
0 
+ 
df. 
¥ 
0 
0 
+ 
ac 
bc 
0 
0 
0 
ac . 
bc 
0 
ad. bd 
ae. be 
bd. 
¥ 
df. 
¥ 
¥ •¥ 
cf .df 
0 
+ 
ab . 
bc 
ad 
bc 
ae. 
bc 
0 
0 
ad. bf 
ae . bf 
df. 
¥ 
ce 
df 
cd 
¥ 
cd . ef 
ce . df 
0 
ab . 
bd 
ac 
bd 
0 
ac . 
¥ 
ae . 
bd 
0 
ae. bf 
+ 
¥ ■ 
¥ 
¥ ■ 
de 
+ 
cd. 
¥ 
+ 
cd. 
¥ 
cf. de 
ab 
bf 
ab 
be 
0 
ac. 
be 
+ 
ac . 
¥ 
ad 
be 
ad. bf 
0 
+ 
ce 
de 
+ 
¥ 
df 
+ 
¥■ 
de 
ce . 
df 
. + 
ce 
df 
+ 
cf . de 
ab 
bd 
0 
ac 
bd 
ac . 
be 
0 
0 
af. bd 
af. be 
ce . 
¥ 
ce . 
df 
cd. 
¥ 
cd. ef 
ce . df 
ad 
bc 
0 
ad 
bc 
0 
+ 
af. 
bc 
ad 
be 
0 
of. be 
de 
¥ 
¥ ■ 
de 
cd. 
¥ 
+ 
cd. 
¥ 
cf. de 
0 
0 
0 
ae 
bc 
+ 
af. 
bc 
ae 
bd 
af. bd 
0 
+ 
¥ ■ 
de 
ce . 
df 
+ 
ce . 
df 
+ 
cf. de 
0 
- 
af. 
bd. 
bc 
¥ 
0 
+ 
ac 
de 
bc 
¥ 
0 
+ 
ad 
ce . 
bd 
¥ 
0 
- 
¥■ ¥ 
ce . de 
ae . 
bc 
af. 
bc 
ac 
bc 
0 
0 
ae . 
be 
af. bf 
0 
¥■ 
de 
be . 
df 
de 
df 
cd. 
df 
cd. de 
ae. 
bd 
af. 
bd 
ad. 
bd 
ae 
be 
<¥■ ¥ 
0 
0 
0 
¥■ 
ce 
be. 
¥ 
ce . 
¥ 
cd 
¥ 
cd. 
ce 
C. X. 
22