128 Mr. G. F. C. Searle on the 



Wo are thus led to regard the momentum as being dis- 

 tributed throughout the field. I£ m denote the amount per 

 unit volume, 



m=G*K/4w) VEH (1(5) 



If 1= jF^so that I is the "impulse" of the applied 

 force, 



i=i^ rrfvEH^^^ (I?) 



§ 13. When the system is set into steady motion from 

 rest, the work done by the applied force F is J ¥udt, where 

 Fu denotes the scalar product of F and u. We have already 

 found that this work is 2W, and thus 



[¥udt=2W (IS) 



Since u is constant, we have 



uI=2W=2(T + U-U ) (19) 



This result is applicable whatever the form of the system ; 

 it is not restricted to systems of revolution. 



As soon as the pulse has got clear of the system, the force 

 F vanishes and the total momentum in the field preserves a 

 steady value. By § 9 we see that the momentum in the 

 pulse tends to a const ant value, which we denote by P, as vt y 

 the radius of the pulse, becomes infinite. The momentum in 

 the new field, which is enclosed by the inner surface of the 

 pulse, also tends, therefore, to a constant value M, which the 

 system has when it is in steady motion. Hence 



I = M + P (20) 



§ 14. In certain cases we can express M, the momentum 

 of the field, for the system in steady motion, in terms of T, 

 the magnetic energy for steady motion. For steady motion 

 we have, as in (3), 



H=KYuE, (21) 



and hence, by (16), 



m = (/*K/47r) YEH 



= (^K 2 /47r) VEVuE 



= (^K 2 /4tt) (u.E 2 -E.Eu) (22) 



Now let 1Ei 1 be the component of E parallel to u and E 2 and 

 E 3 the other rectangular components. Then Eu=E L ?< and 



