53. 



ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL BOUNDED BY 

 TWO SIMILAR AND SIMILARLY SITUATED CONCENTRIC ELLIPSOIDS 

 ON AN EXTERNAL POINT. 



[Abstract.] 



[From the Cambridge Philosophical Society's Proceedings. Vol. n. (1871).] 



No problem has more engaged the attention of mathematicians, or has 

 received a greater variety of elegant solutions, than that of the determi- 

 nation of the attraction of a homogeneous ellipsoid on an external point. 



Poisson's solution, which was presented to the Academy of Sciences in 

 1833, is founded on the decomposition of the ellipsoid into infinitely thin 

 shells bounded by similar surfaces. By a theorem of Newton's, it is known 

 that such a shell exerts no attraction on an internal point, and Poisson 

 proves that its attraction on an external point is in the direction of the 

 axis of the cone which envelopes the shell and has the attracted point 

 for vertex, and that the intensity of the force can be expressed in a finite 

 form, as a function of the coordinates of the attracted point. 



In 1834, Steiner gave, in the 12th volume of Crelle's Journal, a very 

 elegant geometrical proof of Poisson's theorem respecting the direction of 

 the attraction of a shell on an external point. He shews that if the 

 shell be supposed to be divided into pairs of opposite elements with respect 

 to the point in which the axis of the enveloping cone meets the plane 

 of contact, then the resultant of the attraction of each pair of such elements 

 acts in the direction of the axis of the cone, and consequently the attraction 

 of the whole shell acts in the same direction. 



