408 Prof. A. Gray on the Attraction of 



distributions is to apply the theorem o£ Bertrand discussed 

 in § 27. The initial distribution is an elliptic homoeoid, and 

 we know that the family o£ confocal surfaces surrounding it 

 fulfil the condition stated in Bertrand's theorem. For upon 

 any one of them a homoeoidal distribution can be placed so 

 as to produce a constant potential throughout its interior. 

 Hence the confocals are the level surfaces of the field of the 

 original homoeoid. Hence also we can suppose the whole 

 distribution carried out from confocal to confoca], so that at 

 each instant the distribution is homoeoidal on one of the 

 level surfaces, and the path of each particle is along the 

 line of force at the inner extremity of which it was originally 

 situated. 



30. We have found [§ 28, equation (38)] the step of 

 potential from the initial surface to an adjacent one of which 

 the equation is 



2 - ^ k 



a' 2 -f die 

 and it is proved in § 29 that dn=^kdu/p. Thus (38) becomes 



— dY= ,-= - du 



In the same way the step of potential from the level 

 surface 



r 2 



a z + u 



to the adjoining surface for which u has been increased 

 by du is 



/7V— ^ mdu 



~~ = V ? ?"vp?iT)(A 2 + u)(c 2 + « j ' 



Integrating from u = to w=co, and observing that the 

 integral must vanish at infinity', we have 



y = m f" d u ai) 



If in this we insert the value of m stated in (36) it becomes 



V = wpabcdk \ °° , d -- . . (42) 



I£ in (42) we change u to u—u' we obtain 



Y = wpabcdk f * , ^ = - (43) 



Jy >J(a 2 + u-u')(b 2 + u-ii'){c 2 + ii- U ') V } 



