804 jinahjsis of the Mccnn'tqtic Celeste ofM. La Place. 



«iclers in a general manner the attractions of any spheroids ; 

 he recollects in the first place that this attraction is given 

 by an equation of the second order of partial differences. 

 The whole theorv of the attraction of spheroids flows from 

 -this fundamental equation. The author, after making it 

 underao various transformations, undertakes to deduce from 

 it, by means of scries, the value of the function required ; 

 he then shows that with respect to spheroids which differ 

 ven' little from the sphere, we may attain them without 

 the aid of integration, by means of a very remarkable equa- 

 tion which takes place on their surface. It is sufficient for 

 this purpose, to develop their radius in a series of functions 

 of a particular kind, aiven by the nature of the question. 

 The author proves that this development can take place in 

 one way only, and gives shortly afterwards a very simple 

 method of forming it. He afterwards establishes a veiy 

 €lc£ant theoren) relative to the definite integration of double 

 dificrcntials which are the products of two of these func- 

 tions, and he deduces from them, that we may gel rid of 

 the first two terms of the development of the radius of the 

 spheroid, by fixing the origin of the co-ordinates at its 

 centre of gravity, and taking for the sphere, which difltrs 

 very little from it, that which is equal to it in volume. By 

 the help of these ronsiderations, the author obtains, in the 

 simplest manner, the attractions of homogeneous spheroids 

 tlifl'ering little from the sphere on the points which arc 

 either interior or exterior to them; and he extends these 

 results to the case in which the spheroids are heterogeneous, 

 whatever in other respects may be the law according to 

 Nv!,"ich the figure and density of their layers vary. Parsing 

 afterwards to the inquiry respecting the attractions of any 

 given ?|)heroids, which depend equally on the function ex- 

 pressin.!! the sum of their molecules divided by their respec- 

 tive distances from the point attracted, the author show* 

 that this function may be easily determined, when we have 

 its expression in series, for the two cases where the point 

 attracted is situated on the prolongation of the axis of the 

 pole, or in the plane of the equator. This consideration, 

 which simpl'fics considerably the inquiry in question, being 

 applied to the ellipsoid, furnishes a new demonstration of 

 the theorem mentioned above, and which consists in this, 

 that the function which determines the attraction of these 

 bodies is the product of two factors, one of which is the 

 mass itself of the ellipsoid, and the othei only depends on 

 the eccentricities and position of the axes. 

 The author afterwards considers the figure which sphe- 

 roids. 



