Analyfis of the Mccaniqm Celeste of M, La Place. 203 



in ihe interior of an elliptic stratum, whose internal and ex- 

 ternal surraces arc hIiUl-, and similarly situaicd_, is equally 

 atlracied by all parts. 



He alier\\ard:< tibtains the attractions of the spheroid 

 with resp<-ct to the three rectangular axes, by means of a 

 siniile definite integral ; but this integral only being pos- 

 sible in iiself, when the spheroid is one of revolution, the 

 author makes an apphcauon of it to surface^ of this kind, 

 and determines in tinite terms the value of their attractive 

 iorce on a point placed in their interior. 



He afterwards considers the attrnclion of the same sphe- 

 roids on an external pfiint. This incpiiry presents n)ore 

 difficulties than the pieceding, but it may however be re- 

 ferred to it. With this view the author calls to recollec- 

 tion that the attractions of the spheroid, parallel to the three 

 axes, are given iiy the partial ■ dlfl'erences of the function 

 which expresses the t-um of the molecules of the spheroid 

 divided by tiieir respective distances from the points at- 

 tracted. He'oblains the value of this function, when the 

 point attracted is at a very great distance, and he gives an 

 .equation of the second order of partial difi'erences which 

 determine it generally. He afterwards shows, by the help 

 of series, that this function is the product of f.vo factors, 

 one of which is the mass of the s[)hcroid, and the other is 

 merely the function ot its eccentricities, and of the co- 

 ordinates of the point attracted ; whence it follows, that 

 the attractions of two elliptic spheroids which have the 

 same centre, the same position of the axes, and the same 

 eccentricities on the same external point, are to eacli other 

 as the masses of these spheroids. It also follows from',tJ^)is 

 property, that in order to obtain the attractii)n of the pro- 

 posed spheroid on the point attracted, it is sudjcient to 

 knov» the attriction on the saine point of a spheroid, whose 

 eccentricity and position of the axes are the same, and the 

 surlace of which would pass by this point. The author 

 shows that there is but one spheroid which satisfies this 

 condition. The research after the attraction of these sphe- 

 roids in points wlijeh are exterior to them is thus referred 

 to the case where the attracted point is on their surface. 

 Hence rcsuks the expression of this atti action in (iniie 

 terms, when the spheroid i-< an ellipsoid of revolution, 

 which completes the theory of the atiraclion of elliptic 

 spheroids. 'I'he author gives the method of exti ndiiiEi: these 

 results tothecase where the attracting spheroid is composed 

 of variable elliptic layers, ofileiisitv, pu-<uion, and eccen- 

 tricities, according to any given law. J le after. vards con- 

 siders 



