418 VIII. NEWTONIAN POTENTIAL FUNCTION. 



By definition of (3, the parenthesis in the last term vanishes, and 



00 



= 2nabcx 



C d u 



I - 



J (a 8 + w)l/(a 8 + w )( &8 + w )( c8 -|- 



a 



34) ~ = -2xabcy f 



V9 J (6 2 + ^)V(a 2 + ^)(i 



a 



co 



, 



= ZTtaocs 





157. 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 hy the homoeoidal shell without, 

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

 a, &, c, the values for the ellipsoid through #, y y 0, say &a, #&, &c. 

 Since the point is on the surface of this, 6 = 0. 



35) 





o 

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



co 



dV 

 36") -TT = 2rt& abcx 



v$ 



ft, 

 



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



00 



dV 



C du 



I - . 



J (a 2 + ^)]/(^ + ^)(6 2 + ^)(c 2 + ^) 







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

 formula. Integrating with respect to x, y, 0, we have 



00 



A * 



37) V=xabc \l- -^L--^-^ 



ft/ 

 



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

 surface V may be continuous. 



In the case of an internal point the above four integrals may 

 be made to depend upon the first. Calling 



