112, 113] ATTRACTION OF ELLIPSOIDS. 213 



We find for the derivatives of V 



, 



= 7T(lbCfJC 



r du 



I - 



Jo- (a? + u)J(a? + u)(b* + 



, da- 



- TTdbC TT-il -- , ---- 



By definition of <r, the parenthesis in the last term vanishes, 

 and 



du 



fa - <r (a 2 + u) {tf^u) (6 2 + u) (c 2 + u) 



du 



. _ , 



= ^ 



) ' 



113. Internal Point. In the case of an internal point, we 

 pass through it an ellipsoid similar to the given ellipsoid, then by 

 Newton's theorem it is unattracted by the homoaoidal shell with- 

 out, and we may use the above formulae for the attraction, putting 

 for a, b } c, the values for the ellipsoid through #, y, z, say 6a, Ob, Oc. 

 Since the point is on the surface of this, a- 0. 



8F ^ ^ , r du 



'Jo 



o (6 2 a* + u)J(6W + u)(6*b 2 +u} 

 Now let us insert a variable u' proportional to u, u = 6*u, 



r 



Jo 2 (a 2 + ^' 



The 6 divides out, and writing u for the variable of integration 

 dV , r du 



r 

 I 



Jo (a 2 



fa Jo (a 2 + u) s/(a 2 + u) (b 2 + u) (c 2 + u) ' 



So that for any internal point, we put <r = in the general 

 formula. Integrating with respect to oc, y, z, we have 



du 



(a 2 + u) (6 2 + ) (c 2 + u) 



The constant term must be taken as above in order that at the 

 surface V may be continuous. 



