1921-22.] The Faraclay-Tube Theory of Electro-Magnetism. 227 
Then we assume that the volume densities of kinetic and potential 
energy are given by 
U^JED (7) 
T = iHB (8) 
The meaning attached to the above quantities is that if we write 
L- j I I (T-U)dv, 
where the volume integral is extended throughout all space, then L may 
be used as the Lagrangian function in equations of motion of the usual 
form. For the sake of brevity, vectorial general coordinates will be 
employed. In order to preserve the form of the equation 
d 
dt dq dq 
it is sufficient to write, in the case of a vector coordinate r (equivalent 
to the three scalar coordinates a*, y, z), 
d .a , .a ■ a 
ar €,»■ dy dz 
a 
ar 
.a ,.a , / a 
(9) 
This notation in vectorial analysis is of course not generally applicable 
but is convenient for the purposes of the present paper. The general 
results of differentiation which will be required are 
a 
as 
a 
as =a 
as 
Sif/S = 2il/S , 
( 10 ) 
( 11 ) 
where S is any vector variable, a is a constant vector, and j/r is a constant 
self-conjugate linear and vector operator. 
4. To define the general coordinates, let all tubes at a given moment 
be divided into small unit lengths ; and let r be the vector from a fixed 
origin to the centre of one such unit segment, which forms part of a tube 
of the mth set, then the Lagrangian equation corresponding to r will be 
( 12 ) 
,n dv dT 
Now, when a unit length of a tube of the ?>ith set is added to, or 
removed from, an element of volume, the increase or decrease of the whole 
Lagrangian function due to this element will be 
8L = |.^'«d„ 
da. 
'■m 
= -8d, 
,(E + Vq„,B), 
(13) 
