168 Compressible and N on- Homogeneous Masses [CH. vn 



Thus when //, = 0, the equation 



/+*%>- .............................. (436) 



becomes identical with equation (430), which is the equation of the surface 

 of constant density p, where p is connected with q by equation (429). 



The value of V (q), the external potential of a uniform mass of unit 

 density rilling this surface of constant density p, is at once seen, by the 

 methods of Chapter IV, to be 



,...(437), 



q I [(fa* 



where the lower limit of integration // is the root of equation (436) at the 

 external point #, y, z at which the potential is being evaluated. The 

 internal potential is given by precisely the same formula (437) with // put 

 equal to zero. 



169. The formulae for the potential may be simplified by introducing 

 a new variable X equal to p/q 2 . If we further put 



^&- 

 we find that 



(438), 



< 439 >* 



and 



t* 00 /J\ 



(441), 



A' 



where A has its usual meaning [(a 2 + X) (6 2 -f X) (c 2 4- X)J% and the lower limit 

 X' is now a root of 



a 2 



(442) ' 



The same formula (441), with the lower limit put equal to zero, will give 

 the value of the internal potential V { (q). 



170. Having evaluated V 9 (q) and Vi(q), we are in a position to attack 

 equations (431) and (432). Only the second of these equations is of imme- 

 diate importance to- our problem. 



