53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. 419 



Let P' be the point on the ellipsoid E l which corresponds to P on E. 

 Join P'M' and produce it to Q', so that 



M'Q' : P'M' :: MQ : PM. 



Then since M and M' correspond in the above system of points so 

 also do P and P', and the lines joining them both are divided in the 

 same ratio, therefore Q and Q' will be corresponding points in the same 

 system and therefore Q' is also on the ellipsoid E^ 



Now by the property of corresponding points on confocal ellipsoids 



we have 



PM L = P'M' and 



Since the portions of the line PQ intercepted by the shell at P and 

 Q are equal, 



the volumes of elements at P and Q are in the ratio of MP" to MQ*, 

 i.e. are as M'P'" to M'Q'- or as M,P- to Mff; 



therefore the masses of these elements have attractions so that the attraction 

 of the element P on M' = the attraction of the element Q on M', 

 and therefore the resultant attraction of these elements will bisect the 

 angle between Mf and M^Q, i.e. will be in the direction M^M, 



for since MP : M Q : : Mf : M,Q, 



the angle PM& is bisected by MM^. 



Hence the attraction of every such pair of elements will be in the 

 direction M^M, and therefore the resultant attraction of the shell E on M 1 

 is in this direction. 



We have now to find the magnitude of this attraction. 



Let p be the perpendicular on the tangent plane at P, then the 

 thickness of the shell at P is pt. 



Hence if PN be the normal to the surface at P drawn inwards, the 

 elementary surface intercepted by a cone whose solid angle is Sw will be 



Scu. MFsecMPN, 

 therefore the volume of the element is 



,,,, pt.8cj.MP 3 

 pt a> MP" sec MPN= f^D -- *TD\T- 



MP cos MPN 



532 



