116, 117] 



Call 

 then 



therefore 



ATTRACTION OF ELLIPSOIDS. 



sin" 1 (iu) 0, 

 iu = sin 0, Vl + w 2 = cos 0, 

 = cos i sin = Vl -f w 2 + w, 



1'0 = log {Vl + u 2 + u], 

 sin- 1 (iu) = = i log { ViTw 2 + w}. 

 /a 2 - 6 2 



Va 2 + (7 + Va 2 -6 2 ) 



219 



2 - 2 



Hence 



, 



In all these formulae, <r is the larger root of the quadratic 



of f _ 



a 2 + <r + & 2 +<7~ ' 



for an outside point, and a- = for an inside point. In the latter 



ftm^ 



case, we have functions only of the ratio j- . 



117. Energy of Distributions. Gauss's Theorem. If a 



particle of unit mass be at p, (x, y, z) at a distance r from a particle 

 of mass m q , the work necessary to bring the unit particle from an 

 infinite distance against the repulsion of the particle m q will be 



If, instead of a particle of unit mass, we have one of mass m p 

 the work necessary will be m p times as great, 



(2) W pq = ^m p = m p V p =m q V q , 



where F = ^. 



* Thomson and Tait. Natural Philosophy, Part n., 527. 



