58 MR. IVORY ON HOxMOGENEOUS ELLIPSOIDS 



complicated nature of the attractive force, it is difficult to deduce from it whether an 

 equilibrium be possible, or not, in spheroids with three unequal axes. 



The problem is unconnected with the physical conditions of equilibrium: it is 

 purely a geometrical question respecting- a property of certain ellipsoids. 



2. Let the three semi-axes of an ellipsoid be represented by 



K being supposed greater than X' ; and put x, y, z respectively parallel to the axes, 

 for the coordinates drawn from a point in the surface to the principal sections of the 

 solid : from the same point draw the line ^ within the ellipsoid at right angles to its 

 surface ; and ^ being limited by the principal section perpendicular to /<•, the axis of 

 rotation, put j» and q for the coordinates of the end of it in that plane, j) being parallel 

 to y, and q X,o z\ from the condition that ^ is perpendicular to the surface of the 

 ellipsoid, it is easy to deduce the values of;? and ^, viz. 



Again, from the same point in the surface, draw the line ^ in the direction of the 

 resultant of the attraction of the whole mass of the ellipsoid ; and let r and s, respect- 

 ively parallel to y and z^ represent the coordinates of the foot of ^ in the same prin- 

 cipal section as before : then f' will be the diagonal of a parallelopiped of which the 

 three sides are x,y — r, z — s\ and the only three forces acting parallel to the sides 

 of the parallelopiped and equivalent to the single force in the direction of the 

 diagonal, will be proportional to the sides, x, y — r, z — s. Now from the nature of 

 the ellipsoid, the attractive forces perpendicular to the principal sections, are propor- 

 tional to the coordinates x, y, z ; and may be represented by A x,B y, C z : and, as 

 these forces have their resultant in the direction of §', it follows from what has been 

 said, that they will be proportional to x, y — r, z — s. In consequence we have 

 these equations, 



y — r ' z — & 



^ = 3/. (i-x)' '^ = ^- (i-x) = 



and, by combining the values of r and s with those of j9 and q before found, we ob- 

 tain 



r> A 



r —p _ y I + \^ 



S - q ~ T ' Z^ A 



Let <T denote the third side of the triangle which has f and §' for its other sides : 

 then ff will represent the only force which, together with the attractive force §', will 

 produce a resultant in the direction off at right angles to the surface of the ellipsoid. 

 Now ff cannot stand for a centrifugal force unless, in every position, it be invariably 



