﻿Faraday-lube Theory of Electro- Magnetism. 599 



that, again applying (13), we have the change of L due to 

 this cause 



8 s L = d wl v.8r(E + Vq m B), .... (16) 



in which q m is variable (but not d Ml ) . 

 Hence 



8L = 8 1 L + S 2 L = gr[d wV .(E4-Vq Ht B)-V.d w (E + Vq m B)], 



q WJ varying in the first term only, and (L m not at all, and 

 finally 



|^ = d OT v.(E4-^q rw B)- V .^(E + Vq m B) . (17) 



with the same convention. 



In calculating the momentum term ^- we have r = q m . 



o r 



Then by the method employed above in calculating (14), 

 since T is symmetrical in q m and d m , 



£= Vd »* B < 18 > 



This will be the value of ^-r when d m 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 varying in direction and magnitude. 

 It is clear that 



^d OT = d TO v- ( lm, ..... (19) 



since the rate of change of a segment of a straight line, as 

 AD in the figure, will be the relative velocity of its ends 

 (vectorially) ; while, of course, if q m expresses the velocity 

 of any point of the tube, as A, the velocity at D will be 

 (1 + ADV •)<!»*> where AD is the vector element. 

 Thus 



d BL d Trj _ 



zr ^ = t Vd m B 



dt Br dt 



= V(d m v.q»)B + Vd w B + Vd BI (q w 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 TO y • B being of 

 course the term in the rate of change due to motion of the 

 segment with velocity q, H . 



