Ellipsoidal Shells and of Solid Ellipsoids. 403 



(Paris, 1784)*, there is some justification for continuing, as 

 has been very generally done, to associate it, even in its 

 extended form, with the name of Maclaurin. 



23. From (14) in § 16 above we obtain the components of 

 attraction at the point P. The cosine of the angle which the 

 normal at P to the confocal makes with the semi-axis, of 

 which the length is a, is pof/(a 2 + u)k. [See (7) § 13.] 

 Hence for the component X of force in the direction of x 

 increasing on a unit particle at P (14) gives 



X =- 2 ^Fv/^ + u ^ + i()(c , + i( /-o - (29) 



Y and Z are obtained by substituting g/(b 2 -\-u), h/(c 2 + it) for 

 f/(a 2 + u) in this equation. The same results are deducible 

 from (16) by differentiating with respect to u, and multiplying 

 the result by the value of du/df drawn from the equation 



= k 



a~ + u 

 of the confocal. 



In the same way (20) gives 



X=— 2wpabcA 



x s/(a 2 + u) s (b 2 + u) (c 2 + u) 



(30) 



and similar expressions for Y, Z, which may be written down 

 by symmetry. In the case of \ = 0, the factors which 

 multiply /, #, h in these expressions for X, Y, Z are inde- 

 pendent of the values of these coordinates. Hence for an 

 internal point /, g, h of a solid ellipsoid 



X=A/, Y=Bg, Z = C7i .... (31) 



where A, B, C are constants, the values of which are given 

 by (30) and the other two similar equations 



24. I shall now indicate the method which the theory of 

 equivalent distributions affords for the solution of the 

 problem of the ellipsoid. It was shown by Coulomb, for the 

 case of an electrical distribution, that the normal force just 

 outride a closed conductor is proportional to the surface 



* This book is referred to by Todhunter in bis ' History of Attractions,' 

 and be quotes Professor de Morgan as to its rarity. I bave not seen it, 

 and give the reference above only at second band. Tbe University 

 Library has no copy, and though the magnificent edition of Laplace's 

 works,* wbicb is now being published in Paris, bas reached vol. xiii., tbis 

 book bas not been included, tbougb much of later date and ou similar 

 subiects bas alreadv appeared. 



J 2F2 



