Ellipsoidal Shells and of Solid Ellipsoids. 401 



P is the external point and PQ is the normal to the confocal 

 through P to the homoeoid of which a section through PQ is 

 given in fig. 2. A, B are two of the points in which the 

 enveloping'/? one touches the homoeoid, and therefore the line 



AB is the polar of P with respect to the elliptic section made by 

 the plane BPA. The line PQ meets the curve in G and pro- 

 duced again meets it in F. G, F are points in which the line 

 PQ, starting from P and ending in F, is divided harmonically. 

 Similarly, if PE he drawn at rio'ht angles to PG and meet 

 AQB produced in R, the line RQ is also harmonically divided 

 in B and A. It follows that if EQ meet the curve in C 

 and D, EQ is divided harmonically in and D. Thus PQ 

 not onlv bisects the angle APB, but also the angle CPD. 

 Hence DQ/QC = PD/PC. 



Now let a cone of small vertical angle be drawn from Q as 

 vertex with its axis along CD. It will intercept two elements 

 of the homoeoid at C and D, the masses of which are directly 

 as the squares of their distances from Q, while their 

 attractions, per unit of their mass in each case, are inversely 

 as the squares of these distances. Hence the total attractions 

 on a unit particle at Q are equal and opposite. But it has 

 been seen that PD/PC = DQ/QC ; hence the attractions of 

 the same pair of elements on a particle at P must be of equal 

 amount, and being along PD and PC are equally inclined to 

 PQ, and have therefore a resultant along that line. The 

 same thing is true for any other pair of elements intercepted 

 by a cone with vertex at Q, and the whole homoeoid may be 

 exhausted by pairs of elements in this way. Any plane through 

 PQ thus divides the homoeoid into two portions which exert 

 attractions at P equal in amount and equallv inclined to PQ. 



Phil. Mag. S. 6. Yol. 13. No. 76. April 1907. 2 F 



