Note on Surfaces of the Second Order. 323 



of whose sides is the right line SPP', and whose transverse sec- 

 tion, at the distance unity from its vertex, is the indefinitely 

 small area w, the portion PP* of the pyramid will attract the 

 point 8, in the direction of its length, with a force expressed by 

 the quantity 



/I IN 2o> 



--- , U, or ; 

 \P P J r 



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

 the axis of , the attraction in the direction of will be 



2o> cos 



Now, supposing the axis of to be normal to the confocal ellip- 

 soid described through 8, it will be the primary axis of the 

 surface 13, which will be a hyperboloid 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 . There- 

 fore when the force is inversely as the fourth power of the dis- 

 tance, 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.* In the preced- 

 ing year I had obtained the theorems just mentioned, by con- 

 sidering the law of the inverse fourth power ; and, as well as I 

 remember, they were deduced exactly as above, by setting out 

 from the equation (a). But I did not then succeed in applying 



* Memoires de Flnstitut, torn. xiii. 

 Y2 



