431 



This result is useful in questions relating to attraction. 

 For if A be an ellipsoid, every point of which attracts an ex- 

 ternal point S with a force varying inversely as the fourth 

 power of the distance, and if the point S be the vertex of a 

 pyramid, one of whose sides is the right line SPP', and whose 

 transverse section, at the distance unity from its vertex, is the 

 indefinitely small area w, the portion PP' of the pyramid will 

 attract the point S, in the direction of its length, with a force 

 expressed by the quantity 



(\ \\ 2w 



( , w, or — ; 



\p p y r 



and, putting for the angle which the right line SP makes 

 with the axis of %, the attraction in the direction of ? will be 



2wcos0 , - 



• {9) 



r 



Now, supposing the axis of C to be normal to the confocal 

 ellipsoid described through S, it will be the primary axis of 

 the surface B, which will be ahyperboloid of two sheets ; and, 

 the surface being symmetrical round this axis, it is easy to see, 

 from the expression for the elementary attraction, that the 

 whole attraction of the ellipsoid will be in the direction of ?. 

 Therefore, when the force is inversely as the fourth power of 

 the distance, the attraction of an ellipsoid on an external point 

 is normal to the confocal ellipsoid passing through that point. 



Hence we infer, that if u be the sum of the quotients found 

 by dividing every element of the volume of an ellipsoid by the 

 cube of its distance from an external point, the value of u will 

 remain the same, wherever that point is taken on the surface 

 of an ellipsoid confocal with the given one. 



The question of the attraction of an ellipsoid, when the 

 law of force is that of the inverse square of the distance, has 

 been treated by Poisson, in an elegant but very elaborate 

 memoir, presented to the Academy of Sciences in 1833 {Me- 



