in an ellipsoid 359 



or corresponding elements of the two segments are in the ratio of the squares 

 of their distances from P. 



Let us now suppose that A^PA^ is a double cone of an exceedingly small 

 aperture, having its vertex at P; let us also suppose that the density of the 

 redundant fluid at Q l is p lt and at Q 2 is p z ; then since the areas of the sections 

 of the cone at Q 1 and Q 2 are as the squares of the distances from P, and since 

 the lengths of corresponding elements are also, by (5), as the squares of their 

 distances from P, the quantities of fluid in the two corresponding elements 

 at Q 1 and Q 2 are as piQ^P* to p 2 PQ 2 *. If the repulsion is inversely as the th 

 power of the distance, the condition of equilibrium of a particle of the fluid 

 at P under the action of the fluid in the two corresponding elements at Q l 



and<? ' is 



*. ...... (6) 



We have now to show how this condition may be satisfied by one and the 

 same distribution of the fluid when P is any point within an ellipsoid or a 

 sphere. We must therefore express p so that its value is independent of the 

 position of P. 



Transposing equation (i) we find 



i i i i 



' HA ' r>n " T~P ...... (7 1 



QJP 



Multiplying the corresponding members of equations (i) and (7) and omitting 

 the common factor A^P .PA t , 





we may therefore write, instead of equation 6, 



Pi (A 1 Q 1 . Q 1 A ^ = p, (A jQ s 

 Let us now suppose that A^A 2 is a chord of the ellipsoid, whose equation is 



a a 6" c a 

 If we write 



X 2 V 2 Z' 



A2 lir\ 



- 02- p-^2-/ ) > 



then the product of the segments of the chord at (?j is to the product of the 

 segments at Q 2 as the values of p 2 at these points respectively, or 



We may therefore write, instead of equation (9), 



If, therefore, throughout the ellipsoid, 



p = Cp"-*, (14) 



where C is constant, every particle of the fluid within the ellipsoid will be in 

 equilibrium. 



