SUSCEPTIBLE OF AN EQUILIBRIUM. 59 



parallel to the line ^/'ip- + z^ drawn from the point in the surface of the ellipsoid at 

 right angles to the axis of rotation ; and this condition requires that the triangle of 

 which the sides are ff,r — p, s — q, shall be similar to the triangle formed by the 

 parallel lines \/y'^ + z^, ?/, z. From the similarity of the triangles, we deduce 



s — q %' 



and hence, in consequence of the last formula, we finally obtain, 



Every ellipsoid which verifies this formula is capable of an equilibrium when it is 

 made to revolve with a proper angular velocity about the least axis ; for the line ^ 

 representing the attraction upon a point in the surface, the line c will represent a 

 centrifugal force, both in quantity and direction ; and the resultant of these two 

 forces will be perpendicular to the surface of the ellipsoid. 



The equation (1.) results immediately from the investigation of Lagrange, who 

 concluded that it admits of solution only in spheroids of revolution, that is, when 

 X = X' and B = C. By expressing the functions A, B, C in elliptic integrals, M. Ja- 

 coBi has found that the equation may be solved when the three axes have a certain 

 relation. It is therefore demonstrated in general, that a certain class of ellipsoids 

 with three unequal axes is susceptible of an equilibrium on the supposition of a cen- 

 trifugal force ; but it still remains to investigate the precise limits within which this 

 extension of the problem is possible, and to determine the ellipsoid when the centri- 

 fugal force is given. 



3. In order to solve the problem in the view now taken of it, we must have re- 

 course to the equations of Lagrange, which contain all the necessary conditions. 

 Let/ denote the intensity of the centrifugal force at the distance equal to unit from 

 the axis of rotation ; the same force urging the point in the surface of the ellipsoid 

 at the distance \/y^ + z^ from the axis, will be equal tofi^y'^ + z^, the components 

 of which in the directions of 3/ and z are respectively /i/ ?LT\Afz. Now 



.Ax, By, C z, 



are the attractions of the mass of the ellipsoid ; wherefore the total forces urging the 

 point in the surface are 



A^, (B-/)i/, {C-f)z. 



These forces must have their resultant in the direction off perpendicular to the sur- 

 face of the ellipsoid ; and as they are parallel to the sides of a parallelopiped, of which 

 ^ is the diagonal, they will be proportional to those sides, that is, to 



I 2 



