128 PROCEEDINGS OF THE AMERICAN ACADEMY. 



usual definitions of field and energy the law becomes merely a geometric 

 theorem in four dimensional non-Euclidean geometry. It is reason- 

 able, however, to suppose that the use of the general form of the law 

 could not remove the difficulty; for the point d'arret arises under 

 conditions of relatively minimal velocity. Moreover, a qualitative 

 discussion of the equation which arises under the general law indicates 17 

 that the difficulty is not removed. 



It is tolerably clear that we cannot avoid the point d'arret and main- 

 tain simultaneously the point electron, ordinary mechanics, and ordi- 

 nary electromagnetic theory. This conclusion may be welcome to the 

 school that follows Bohr in abandoning both mechanics and electro- 

 magnetic theory 18 as hitherto understood even when transformed as 

 suggested by the principle of relativity. And we certainly do not wish 

 to imply that this abandon may not work out satisfactorily. 



The conservative, however, will naturally try to work out of the 

 present difficulty by abandoning the point electron, especially as elec- 

 trons are generally supposed to have magnitude. It is a bit hard to see 

 why the assumption of a very small finite size for the electron must 

 fundamentally vitiate the reasoning which leads to the law of the square 

 of the acceleration; but we shall not go into this question, because we 

 are not acquainted with any derivation of this law which does not in 

 some form practically assume the electron to be a point. Moreover, 

 the work of Lorentz and Ritz, starting with a distributed electron and 

 introducing approximations based on the assumption of small velocities 

 and accelerations, establishes a reaction proportional to the rate of 

 change of the acceleration and thus leads on mechanical principles to 

 a law of radiation (at any rate to a loss of mechanical energy) propor- 

 tional to the (scalar) product of the velocity and rate of change of accele- 

 ration. And it is interesting to note that this reaction and rate of 

 change of energy is actually independent of the size or shape assumed 

 for the electron, that is, may be assumed to hold for the point electron. 



Howeverso apologetic one may be in regard to (16), that equation 

 seems somewhat less in need of apologies than (15) or its derivation 

 from (15) in integrated form. 



Massachusetts Institute of Technology, 

 Boston, Mass, May, 1914. 



17 Just what assumptions in non-Newtonian mechanics should be made to 

 treat this problem it is perhaps difficult to state, and consequently the question 

 is here left as a qualitative judgment of the author. 



18 N. Bohr, Phil. Mag., (6) 26 (1913) a series of three articles On the Constitu- 

 tion of Atoms and Molecules, pp. 1, 476, 857. 



