Difficulty in Theory of Attraction of Spheroids. 35 r 



the nature of the surface or of the magnitude of the sphere ; the 

 only things supposed being that all the lines drawn from P meet 

 the surface again but once, and that no part of it passes beyond 

 a plane through P at right angles to PQ. 



"With respect to the limit of the quantity F", it is obvious 

 that if a hemisphere be described from P" as a centre, with a 

 radius equal to the greatest difference 8 between the lines Pp, 

 P'p'y the solid P"p" will lie wholly within this hemisphere, and 

 consequently V" will be less than the value of V for the hemi- 

 sphere, that is, less than ?rS 2 ; for here all the little pyramids 

 from the centre have the same length S, and their bases are 

 spread over the hemispherical surface ; wherefore V" = 2w 

 x |-S 2 = TrS 2 . All this is independent of anything but the suppo- 

 sition just mentioned. 



If now PQ be supposed to be a spheroid of any sort, 

 slightly differing from the sphere P'Q', and such that the 

 line PQ, perpendicular to the surface at P, passes nearly 

 through the centre, then all the differences, of which S is the 

 greatest, being of the first order, the quantity V", which is less 

 than TrS 2 , will be of the second order ; and therefore neglecting, 

 as Laplace has done, the quantities of that order, we get the 

 theorem in question. 



It may be well to apply the general theorem to the simple 

 case in which the first solid is a sphere of the radius ', because 

 both Lagrange and Ivory have used this case to show that the 

 reasonings of Laplace are incorrect. In this instance, then, the 

 surface described by the point p" is that of a sphere whose 



radius is the difference between a and a' ; and the values of 



4,44 4 



F, F', V", and A, are -^Tra'*, -^ira?, - ?r (a' - a) z and ^ira' respec- 

 o o o o 



tively. 



Substituting these values in the equation F+ V - V" = 2aA, 



4 

 and omitting the common factor ^TT, the resulting equation 



o 



a" 1 + a 2 - (a' - a) 2 = 2aa' 

 ought to be identical ; and so it manifestly is. 



November, 1831. 



