544 PROCEEDINGS OF THE AMERICAN ACADEMY. 



dC .r-^L' E 



-dt-^^L-'^-V ^*> 



where L' = dL/dt, and if we make use of the usual notation,^ and put 

 P = (r + L')/L, Q = E/L, we shall have in general 



(7=e-p(M+ rQe'dt), (5) 



Jo 



where j, = £p.,, = ^(1) + £f. (6) 



That is, if < ^ < T, 



Q^fe^o L^c^j^.^ eJ^^-dtX (7) 



and, in particular, 



T PTrdt p pT ftTdt 



C^ = ^e~-Jo L{C, + ^ ehL.dt). (8) 



In this expression L is to progress always in the same direction from 

 Zq to Xi, and cannot pass through the value zero, so that the limit of 

 Ct as T approaches zero has the familiar value 



Limit Ct = ^, (9) 



which might have been found directly by integrating (2) with respect 

 to t from to J"; the electromagnetic momentum has no sudden 

 change. Equation (9) follows immediately, of course, when one makes 

 use of the usual analogies between the phenomena of ordinary me- 

 chanics and those of electromagnetism. Equation (2) is in form like 

 the equation of motion of a system the mass of which changes with 

 the time in a certain given manner and which is under the action of 

 a constant accelerating force and a retarding force proportional to the 

 velocity. Let a moving mass L grow steadily during its motion by 

 the gradual accretion of small particles which, originally at rest, are 

 suddenly made part of the moving system, much as the links of a fine 

 chain which has been lying on a table are successively set in motion 

 when one end of the chain is lifted more and more ; or let the mass 

 L decrease steadily by the loss of small particles each of which leaves 



* Forsyth, Treatise on Differential Equations, § 14. 



