MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 147 



Strictly, M and k are functions of the frequency, but with this limitation we obtain as 

 an integral 



On average for a periodic motion the term kxx disappears, and we get the equation 

 of balance of energy, for the mean value of k\x 2 dt is the quantity of radiation as 

 calculated by LABMOR. 



It is clear that we cannot reverse this process, and that it breaks down when the 

 motion is not periodic. 



A similar objection applies to the more elaborate expansion by SOM MERFELD 

 (' Go'tt. Nachrichten,' p. 410, 1904) for the reaction in powers of the acceleration. 



ABRAHAM ('Electrician,' p. 868, 1904) has given a still more general formula for 

 the reaction. Apart from difficulties as to the distribution of the charge on the 

 particle, his expression does not enable one to determine the important question as to 

 what kind of motion is really possible. Many of the calculations I have seen either 

 ignore the surface conditions or introduce assumptions about rigid electrification 

 which seriously detract from the value of the conclusions. 



Experimental work of recent years has naturally directed attention to the problem 

 of the dependence of electrical inertia on the speed. Since the problem of accelerated 

 motion has not hitherto been solved, extensive use has been made of the solution for 

 steady motion. The process of deriving an expression for the electrical inertia from 

 the expression for the energy of the steady motion has given rise to ambiguity of 

 meaning which is inevitable with such a method, and involves a serious fallacy of 

 dy nan i i cal re ason ing. 



It will be generally admitted that if we introduce steady motion values in a proper 

 Lagrangean energy function, and then apply the usual methods, we have no right to 

 expect correct results. This fallacy is shown by the example ij = >' '(, thus 

 (h//i/x = I for all the values of x if we first differentiate, but if we put x = n and 

 then differentiate we get dyjdx = 0. 



But, apart from this, a fallacy is involved. If the energy function lias been derived 

 from a Newtonian system of equations and the kinetic energy involves squares of the 

 velocities, inertia may be defined from the energy function in a variety of ways, each 

 of which gives the same result. Thus, if 



K.E. = T = Jwm 2 , 

 we may define mass as 



2T 1 d'Y d T dT 



2 ' 7 ' " 7 ' *^ 7 ~~v ' 



u u au du ii, du 



We may devise an infinite number of definitions, all of which are consistent as 

 long as T = fyniu*. 



If, however, we find by any process that the kinetic energy involves higher powers 



u 2 



