( 348 ) 



in case of in finite velocity ; its \aine has been calculated, for the 

 first time, as far as I know, in my second paper. Further you derive 

 from the same formulae tlie surprising result, stated in my second 

 paper not only for the case of stationary, rectilineal motion, but for 

 any motion yon like : The motion of a surface-charge \vith a >'elocity 

 exceeding that of light, is actually impossible, requiring continually 

 an infinite supply of force. In order to make this more evident, 

 let me point out : the more the charge is concentrated, the more the 

 force will increase ; in case the charge is concentrated at one point, 

 the force is infinite even in I lie case of a velocity less than that 

 of light. 



It may seem unsatisfactory, to l^e restricted to the special shape 

 of a sphere. Only a few of the following results are independent 

 of this restriction, that is those, \vhich do not contain the radius of 

 the electron, e.g. the approximate formulae for the field of a charge, 

 stationai'ily moved, in the case of a velocity less than that of light, 

 and those in case of a velocity exceeding that of light, in the regions 

 I and III (§ 4), whereas the formulae relating to the limit of tlie 

 shadow of motion, that is to the region II, depend on the special 

 spherical shape. Yet it is evident, that on moi'e general suppositions, 

 you could probably not proceed so far. 



It is kown, that H. A. Lokentz^) has lately supported the hypo- 

 thesis, that the shape of the electron is variable, conforming itself 

 to a "HEAVLSiDE-ellipsoid", according to its momentary velocity. As 

 for velocity exceeding that of light tiiis hypothesis fails, because in 

 this case you can hardly speak of a "HEAVisiDE-hyperboloid". So I 

 have not been able, to use this hj'pothesis. 



§ 2. Green's TJieorem. 



All natural philosophy proves the wonderful power of Green's 

 theorem. We shall use it here very much like Kikchhoff") in his 

 enunciation of Huygens' principle. 



Let (p be the scalar potential, satisfying the differential equation : 



1 ö> 



JLw Z= r p, (1) 



where c means tlie velocity of light, and q the density of charge of the 

 electron; as regards the choice of units see H. A. Lorextz, Ency- 

 klopiidie der Mathem. Wissenschaften Bd. Y. Art. 13, N". 7. 



1) K. Akademie van Wetenscliappen te Amsterdam Mei 1904. Prociicdings p. 80'). 



2) Vorlesungen über Mathematische Optik, 2te Vorlesung, § 1. Leipzig 1891. 



