350 Difficulty in Theory of Attraction of Spheroids. 



terminated by the surfaces of the solid and of the sphere, the 

 third, Py, i n ^ ne same direction with them and equal to their 

 difference, without regarding which of them is the greater, and 



suppose all the points p" 9 taken according to the same law, to 

 trace the surface of a third solid. Let Pp, P'p'> P"p" > be edges 

 of three small pyramids with their other edges proceeding from 

 P, P', P", parallel, and having of course the same solid angle, 

 which we shall call $, and denote by r, r', r", their respective 

 lengths, and by F, F', F", the values of the function Y for each 

 of them. Drawing pR perpendicular to PQ, the attraction of the 

 pyramid Pp in the direction of PQ will be equal to x PR. 

 Call this attraction A, and let a be the radius of the sphere. 



Since /' is the difference of r and r', we have r 2 + r' 2 - r" 2 

 = 2rr' = 2PM x P'Q', and multiplying by 1 </> we find 0r 

 + r ' _ !0 r "2 = 2a<j> x PR, that is F+ V - V" = 2aA. The 

 same thing is true for any other three pyramids similarly related 

 to each other, throughout the whole extent of the three solids 

 which are exhausted by them at the same time ; and hence, if 

 we now denote by F, F', F", the whole values of the function Y 

 for the three solids, and by A the whole attraction, of the first of 

 them parallel to PQ on a point at P, we shall still have F+ V 

 - V" = 2aA. 



To express this general theorem in the notation of Laplace, 

 we have merely to observe that the attraction A is synonymous 



with - ( ), and that the quantity V for the sphere is equal 



\U')" J 



4 



to ira z . Substituting these values, we find 



an exact equation, differing from the approximate one of Laplace 

 only in containing the quantity V ', and totally independent of 



