522 
PROFESSOR H. LAMB ON ELECTRICAL 
their first derivatives must be continuous. This follows at once from the expressions 
for F, G, H in terms of the electric currents in the field, viz., 
F=|| j V —dxdy'dz 
G=| j | V ~dx'dy'dz > 
H = || ^—dxdy'dz' 
( 5 )> 
where r denotes the distance from the element dxdy’dz ' to the point (x, y, z) at which 
the values of F, G, H are required. Hence if a, b, c be the components of magnetic 
induction, viz., 
b = 
dF 
dz 
dG" 
dz 
d H 
dx 
(«), 
<H> <tF 
dx dy 
these quantities will be continuous at the surface of a conductor. Conversely we may 
show that if F, G, H, a, b, c be continuous then the first derivatives of F, G, H will 
all be continuous. For this it is sufficient to prove that their derivatives in the 
direction of the normal will be continuous. If l, m, n be the direction-cosines of the 
normal, w r e have 
7 dF , dF , dF 7 dF , cttl , dFt , 7 
v —7 -— \-nb— me 
dx dy dz dx dx dx 
dG 7 dG 
( 7 ), 
by (2), and it is easily seen from geometrical considerations that the continuity of G 
implies the continuity of (m.dG/dx — l.dG/dy), and so on. Hence if F, G, H, a, b, c 
be continuous, the first member of (7), and the corresponding expressions for the 
normal derivatives of G and H, are continuous. 
From this point the letters u, v, iv wull be used to denote solely the components of 
current in the conductors. The components of current in the dielectric are f g, h. 
The general solenoidal conditions to be satisfied by u, v, w andy^ g, h, viz., 
du dv dw 
dx'dy dz 
and 
(8), 
