160 
ON THE DOUBLE ©-FUNCTIONS IN CONNEXION 
[662 
[5] 
(ft 
Y> 
ft 
«)(Z, 
ft 
ft 
IF) = 
= 0, 
[6] 
(y, 
-8, 
a, 
-£)(Z, 
ft 
ft 
W) = 
= 0, 
[7] 
(y> 
8, 
a, 
£)(Z, 
ft 
ft 
W) = 
= 0, 
[8] 
(ft 
- Y. 
ft 
- a)(X, 
ft 
ft 
W) = 
= 0, 
[9] 
(ft 
a, 
8, 
Y>(Z, 
ft 
ft 
W) = 
= 0, 
[10] 
(a, 
-ft 
Y> 
- 8)(X, 
ft 
ft 
W) = 
= 0, 
[H] 
(«> 
ft 
Y> 
8) (Z, 
ft 
ft 
W ) = 
= 0, 
[12] 
(ft 
- a, 
8, 
- y) (Z, 
ft 
ft 
W) = 
= 0, 
[13] 
(8, 
Y> 
-ft 
- a) (X, 
ft 
ft 
W) = 
= 0, 
[14] 
(y» 
-8, 
- a, 
ft)(Z, 
ft 
ft 
W) = 
= 0, 
[15] 
(y, 
ft 
- a, 
-ft)(z, 
ft 
ft 
W) = 
= 0, 
[16] 
(5, 
- Y> 
-ft 
«)(Z, 
ft 
ft 
W) = 
= 0. 
I repeat in a new order the sets of coefficients which belong to the several 
squares, viz. these are 
(1) P 2 (a, -ft - 7, 8), 
(2) Q 2 (ft a, -8, - 7), 
(3) Q 2 (ft -a, -8, 7 ), 
(4) P x 2 (a, ft - 7, - 8), 
(5) P 2 2 (7, - 8, a, - ft, 
(6) ft 2 (8, 7, ft a), 
(7) ft 2 (8,-7, ft - a), 
(8) P 3 2 (y> ft A 
(9) P 2 2 (a, -ft 7, - 8), 
(10) Q 3 2 (ft a, 8, 7), 
(11) Q 2 (ft -a, 8, -7), 
(12) P 3 2 (a, ft 7, ft, 
(13) P 2 (7, -8, -a, ft, 
(14) ft 2 (8, 7, _ft _ «), 
(15) $ 2 (8, -7, -ft a), 
(16) Pi 2 (7, 8, -a, -ft. 
And I remark that, if we connect these with the multipliers (F, — X, W, — ft), we 
obtain, except that there is sometimes a reversal of all the signs, the same linear 
functions of (X, F, Z, W) as are written down under the same numbers in square 
brackets above: thus (1) gives 
(a, -ft -7, ft (ft -X, W, - Z), which is (ft «, -8, -y)(X, F, ft W), = [1];