1813.] 



Imperial Institute of France, 



151 



for points situated in the prolongation of the tliree axes might be 

 extended to all the points situated in the plane of any one of the 

 three principal sections of the solid: and considering the problem 

 in all its generality, he showed that the difficulties of integration 

 might be overcome ; but he acknowledges, at the same time, 

 that the analysis was exceedingly complicated. 



M. Biot, by a happy complication of a theorem of M. 

 Legendre with a complete integral upon which M. le Comte 

 Lagrange had founded his theory of fluids, succeeded afterwards 

 in reducing the attraction of an elipsoid upon an exterior point 

 to the case when this point is situated upon the surface of the 

 elipsoid itself. 



After so many efforts, in which all the resources furnished by 

 the most skilful analysis seems to have been exhausted, little 

 hopes were entertained of obtaining a more easy solution. This 

 however, has been accomplished by Mr Ivory, who, by a trans- 

 formation as simple as ingenious, has demonstrated that the 

 attraction of a homogeneous elipsoid upon any external point 

 whatever may be reduced to tliat of a second elipsoid upon a 

 point within it. " Tlius," says M. Legendre, " the difficulties 

 of analysis which the problem exhibited disappear at once ; and 

 a theory which belonged to the most abstruse part of mathematics 

 may now be explained in all its generality in a manner almost 

 entirely elementary." The object of M. Legendre, in the new 

 memoir which he has communicated to the Class, is to take 

 advantage of the discovery of Mr. Ivory to present the entire 

 theory of the attraction of homogeneous elipsoids in all the 

 simplicity of which it has become susceptible. 



He states, first, the general formulas of the problem which 

 extend to all the points within and without the elipsoid. He 

 then explains, in a very clear manner, the method of Mr, Ivory, 

 which consists in making the surface of a second elipsoid pass 

 through the external point. The principal sections of this second 

 elipsoid are situated in the same planes, and described from the 

 same foci, as the corresponding sections of the given elipse. 

 Then upon the surface of the first elipsoid a point is taken, such 

 that each of its co-ordinates is to the corresponding ordinate of 

 the exterior point, in the same ratio as the analogous semiaxes of 

 the two elipsoids. The point thus chosen w^ill be within the 

 second elipsoid, and we may calculate its attraction parallel to 

 each of the three axes of the second elipsoid. In order to adduce 

 the three attractions of the external point to the first elipsoid, it 

 is only necessary to multiply those of the second by the ratios 

 between the products of the two other axes in the two elipsoids. 

 Hence it is obvious that by a simple multiplication, the second 

 case of the general problem, considered hitherto as extremely 

 difficult, is reduced to the first case, already completely and 



