198 Wisconsin Academy of Sciences, Arts and Letters. 



confocal ellipsoids. In case, however, the outside particle 

 is in the plane of the equator or line of the poles, it is an 

 evident inference from my investigation that two confocal 

 ellipsoids having their foci at the same points, attract the 

 same particle outside of both, as their masses. In case the 

 outside particle is otherwise located, I have a short and 

 simple demonstration which, if put in at the end of Art. 18 

 of this paper, would prove that confocal ellipsoids attract 

 any outside particle, as their masses. Todhunter in the 

 second volume of his History of the Theories of Attraction 

 and the Figure of the jE'ar^/i, says, referring to the expedient 

 of confocal ellipsoids: "Legendre we see arrived at his 

 theorem incidentally as he was developing a new demon- 

 stration of Laplace's theorem; and the improvement subse- 

 quently effected by Ivory in the treatment of Laplace's 

 theorem has probably much diminished the interest which 

 would otherwise have continued to belong to Legendre's. 

 Nevertheless it is to be wished that a simple investigation 

 could be supplied of the remarkable result; and perhaps this 

 may be attained in consequence of thus drawing attention 

 to it." 



It is true that the final results of the investigation of 

 this paper reduce to the same as those well known by ma- 

 thematicians. It is also doubtless true that the modern 

 method of the Calculus is the best for obtaining a specific 

 conclusion. But it seems to me that a new and true geo- 

 metrical investigation on an intricate subject like this, must 

 furnish another lamp by which can be seen new deductive 

 truths. However this may be, I feel well compensated for 

 the brain force expended in the general mental culture re- 

 ceived, and especially in a sharpened ability to detect the 

 fallacious logic in my own scientific and philosophic 

 thought. 



2. Two particles at a sensible distance apart attract each 

 other with a force directly proportional to the product of 

 their masses and inversely proportional to the square of 

 their distance. This law is accepted because it is the only 

 law of attraction that accords with physical phenomena. 



