308 
Proceedings of the Royal Society of Edinburgh. [Sess. 
of two directed spheres with Lie co-ordinates (x v y 1 ,z v Ct 1 ), ( x 2 ,y 2 ,z 2 ,ct 2 ) 
passes through the origin. If the origin of co-ordinates lies within the 
second of these spheres, the common tangent cone cannot pass through the 
origin unless the cone is imaginary and has its vertex at the origin. Hence, 
if xl + yl + zl<cH\, the only singularities of the field occur when x v y v z v t 1 
satisfy the equations 
x d'2 ~ X fl = Vl^2 ~ ” O’ — ~ 
If we regard (x v y v z v tf) as current co-ordinates, our field can he inter- 
preted as the electromagnetic field due to an electric pole which passes 
through the point x 2 , y 2 , z 2 at time t 2 and moves with uniform velocity less 
than c along a straight line through the origin. The interpretation when 
(x 2 , y 2 , z 2 , t 2 ) are taken as current co-ordinates is similar. The partial 
symmetry of the expressions for the vectors E and H in the two sets of 
co-ordinates (x v y v z v tf), (x 2 , y 2 , z 2 , t 2 ) is worthy of note. 
§ 5. One of the theorems of § 4 may be generalised as follows : — 
If a complex vector M = H + i E satisfies the system of partial differ- 
ential equations 
diVpM=0 (2? = 1,2, . . . n) 
c dtp 
and possesses continuous second derivatives , it is a right-handed multiple 
wave- function* This is easily verified by differentiation. 
A solution of the above system of equations is given by 
M 4>(£, g 2 , ... $ n ; y] ± , rj 2 , . . . rj n ; 6)C(6)d6 . . . (9) 
where f p , rj P) C (0) have the same meaning as in § 4. A solution may also 
be obtained by writing 
i 0 L 
M = i rotXg=- —fi + grad^Ag , 
c dtp 
L q = - tr + i rot ^ + g ra( b K 5 A q = - diVgG - - “ , 
C 01 q C dtq 
where the scalar K and each component of the vector G is a right-handed 
multiple wave-function. If we write 
rvr. 
0 = J o G*(^1 , £ 2 , ■. . . tn ; Vl > V2 » • ■ • Vn } @)d6, 
f%TT 
H = j o ,4, • • • 4; Vi 5 V2 > • • • Vn > 0)d0, 
* This means that the equations div p grad 2 Y = — fiff } 
C 2 dtpdtq 
c rot„ gradg V = i grad. 
dt c 
0y 
— t'gradg ~ (p, q = l, 2, . . . n) are satisfied by each component of M 
dtp 
