312 
Proceedings of the Royal Society of Edinburgh. [Sess. 
To verify this we may first generalise Whittaker’s expression * for 
a wave-function and obtain the following expression for a multiple wave- 
function which is completely neutral : 
fir f2ir 
V = f I F(a 15 a 2 , ... a n ) 6, cf))d6d<f> 
where 
a v = x p sin 6 cos <£ + y p sin 0 sin <f> + z p cos 6 - d p . 
Substituting the expressions for x p , y P , z p , t v , we obtain an expression 
of the form 
where 
Y = 
<3>(a, /3, y, S, 0, cfi)d6dcfi 
a = x sin 6 cos <j> + y sin 6 sin <£ + z cos 6 - d , 
/3 = x sin 6 sin <f> — y sin 6 cos cf> - iz + id cos 6 , 
y = x cos 0 -J- iy — z sin 0 cos - id sin 0 sin c fi, 
8 = x + iy cos 0 - iz sin 0 sin cf> - d sin 0 cos <f>. 
It is easy to verify that the above expression is a wave-function. If 
we regard (x, y, z, ict) as the retangular co-ordinates of a point in a space 
of four dimensions, the formulae (11) represent a particular type of trans- 
formation of rectangular axes. 
* Math. Ann., Bd. lvii (1903). 
{Issued separately March 1, 1917.) 
