53] ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. 415 



About three years later, M. Chasles shewed that Poisson's solution 

 might be greatly simplified by the consideration that the axis of the 

 enveloping cone is identical with the normal to the ellipsoid which passes 

 through the attracted point and is confocal with the exterior surface of the 

 shell. 



This mode of enunciating the direction of the attraction has the 

 advantage of making known the level surfaces with respect to the attrac- 

 tion of the shell on external points. 



In 1838, M. Chasles presented to the Academy of Sciences a very 



simple and elegant investigation, in which he arrives at Poisson's results 



respecting the attraction of a shell on an external point, by a purely 

 synthetical method. 



M. Chasles' method is founded on Ivory's well-known property of cor- 

 responding points on two confocal ellipsoids, and on some elementary 

 propositions in the theory of the Potential. 



Struck by the simplicity and beauty of Steiner's method of finding 

 the direction of the attraction of a shell on an external point, the author 

 of the present paper was induced to think that by means of the same 

 method of decomposing the shell into pairs of elements employed by Steiner, 

 a correspondingly simple mode of determining the intensity of the attraction 

 might probably be found. The author has been fortunate enough to succeed 

 in realizing this idea, and the result is the method contained in the first 

 part of the present paper. 



This method is throughout quite elementary. It requires the knowledge 

 of only the most simple properties of ellipsoids, including Ivory's well-known 

 property respecting corresponding points on two confocal ellipsoids. 



The proof of the theorem respecting the direction of the attraction 

 differs from that given by Steiner, and harmonizes better with the method 

 employed for determining the intensity of the force. No use is made in 

 this method of the properties of the Potential. 



The second part of the present paper is devoted to what the author 

 considers to be an impi'ovement on M. Chasles' method of determining the 

 attraction of a shell on an external point. Its novelty consists in the mode 

 in which the intensity of the attraction of the shell is found. M. Chasles 

 first compares the attractions of two confocal shells on the same external 

 point. He then takes the outer surface of one of these shells to pass 



