JEolotropic Potential. 449 



the multiplier 1 — r 2 (or &), and so vanishes, Avhile T has not 

 this multiplier and becomes infinite. Thus it may fairly be 

 concluded that electrical action will not generate a relative 

 velocity attaining that of light. The electric forces for this 



case are given by X'= -^L,/-,... ; for the interior in the limit 



L = M =1, and N becomes infinite as <£ . Thus electric 

 force vanishes with k, but the ^-component is great in relation 

 to the others. The magnetic force is given by l'oc = <j r L' — i-Y', 



and for the limit u= — -//, j3=§ } .v } 7 = in the interior, 

 /'. r. the force is finite. 



A- regards the character of the surfaces 



+ ( .2 + \(i-r 2 ) =1 



when /• approaches 1, the expansion of the c axis with 

 increase of X is reduced, and in the limit all the surfaces are 

 crowded in contact with the ends of this axis, while expanding 

 freely in the equatorial plane ; and this accords with the 

 relatively great value of the component Z' of the force. 



If r>l, the integration for <£ starts from X = as before, 

 this being necessary to determine the constant at the surface; 

 but it proceeds only to a critical value X = c 2 /(V 2 — 1). Since 

 when /• is only slightly greater than 1, the above critical value 

 is infinite, this course of integration establishes continuity 

 with the case r<l- 



The value of O for the sphere then differs only from the 



case /•< 1 in having log - — - in lieu of log 1 -: thus if 



the solution were applied beyond r==l, S would become 

 negative through the factor l — r'\ but T would only suffer 

 the change in the form of the logarithm. 



But the case of volume-distribution breaks down through 

 the breach of ( ( J1) and (93). In the latter, where an inte- 

 gration of energy through the volume was in question, \/\/j 

 - required to vanish at the upper limit, which with r< 1 

 was assured by the cube of X contained in J; but when r>l, 

 one factor in J is c 2 — X(r* — 1), and the vanishing o£ this 

 factor for the upper limit makes X \/j infinite. As regards 

 '•'1 The details for the spheroid need consideration. The 

 values of ^qLoMo in (110) (ii.) are correct for this case if 



is written for log - — ; but lor N fl the upper limit 



7 — C — y L L 



of integration introduces an infinite constant. N has in 



