310 Proceedings of the Royal Society of Edinburgh. [Sess. 
use S to denote the vector with components (sin 6, — cos 0, 0), we find that 
where 
, 4 . • • 
• £n j 
< Vn V2> 
••• In) 
6)d8 
• • • 
L- 
'Vi* 
■ • * Vn j 
6%6 
rcA*i 
+ i 
[s— 1 
+ C®* 
+ s 8 A* 
L J 
- d Vq J 
34 
°v, 
(c dG 
*\_ 
Ys 0G *' 
\ 0K* 
\ C Ti 
J 
V d V q, 
) 3 Vq ' 
L*= S A* + 
°Vq 
A*: 
It should be noticed that 
(CL*) = 0, (SL*) + A* = 0, 
[C L*] = i C A*, i[ SL*] = L* + SA* 
and that if L* and A* satisfy these conditions then equations (10) are 
satisfied. 
The question now arises whether the parts of L* and A* depending on 
the vector G* can be expressed in terms of a scalar of type K*. This is 
not generally the case, because if we had 
0G* 
d Vq 
+ i 
k 0G ■ 
'd£ a 
+ i 
.0G' 
Q 0G*' 
L S d Vq _ 
.9G 
= C ^° + S^° 
0K C 
we should find on eliminating K° that 
0 2 G* 
3 Vq 2 
or 
+ i 
j 8 2 G* 
_ 3 g drj, 
,8 2 G' 
J + [ s ?] - <°Ss) - s ( s ?) - + 01 8 
0 2 G* 
343^ 2 
k 0 2 G* 
'W 
+ 2 S 
9 2 G* 
’343^ 
- C 
^0 2 G* 
3^ 2 
= 0, 
where C is the vector with components (cos 6, sin 0, —i). This last 
equation is not generally satisfied, and so the above statement is 
justified. 
§ 6. We shall now consider the question whether it is possible to find a 
set of functions 
x p = X p( x > y, t), y v = Y p (x, y, z, t), z p = Z p (x, y , z, t), t p = T p (x, y, z, t) 
p = l, 2, . . . n 
such that a multiple wave-function of the n sets of four quantities 
(x p , y p , Zp, tp) is an ordinary wave-function when considered as a function 
of x, y, z, and t. A partial answer to this question has already been given,* 
* Bulletin of the American Mathematical Society , April (1916). 
