1921-22.] The Faraday-Tube Theory of Electro-Magnetism. 229 
varying in the first term only, and not at all, and finally 
^^^ = d„v.(E + Vq„B)-v.d„(E + Vq„B) . . . (17) 
with the same convention. 
0L 
In calculating the momentum term ^ we have Then by the 
method employed above in calculating (14), since T is symmetrical in 
Q.m and d^ , 
If = Vd„B (18) 
cqm 
This will be the value o£ when d,j^ is a unit length of tube, but in 
performing the complete differentiation to time in (7) we must remember 
that any length of tube will in general be continually vaiying in direction 
and mao;nitude. It is clear that 
7 , . q^j^ , . . . . . ( 1 9 ) 
(it 
since the rate of change of a segment of a straight line, as AI) in the figure, 
will be the relative velocity of its ends (vectorially) ; while, of course, if q^ 
expresses the velocity of any point of the tube, as A, the velocity at D 
will be (1 -f ADv • )qm ? where AD is the vector element. 
Thus 
(it di 
= V(d,„v.qm)B + Vd[„,B + Vd,^(q,^V.B),. . . (20) 
where B is the rate of change of B at a fixed point coincident with the 
moving centre of the segment, q^v • B being of course the term in the rate 
of change due to motion of the segment with velocity q^. 
Equation (12) is therefore by (17) and (20), 
V(d,7,v . qm)B + Vd^^B - Vd^(q^v . B) 
— ■ (E-f Vq^^B) + V • dm(E + Vq^B) = 0 , . . . (2!) 
d,„ and q^ji being constant in the last term, and y operating forwards only. 
In carrying out the simplifying transformations we may drop for the 
moment the suffix "/n. 
From the last two terms we have, in part, 
- d V . E + V . dE = + Vd V vE 
= + Vd curl E 
( 22 ) 
