35 



one plane. This determination obviously comprehends all spheroids 

 of revolution ; but, on account of the complicated nature of the at- 

 tractive force, it is difficult to deduce from it whether an equilibrium 

 be possible or not in spheroids of three unequal axes, a problem vv^hich 

 is unconnected with the physical conditions of equilibrium, and 

 which is a purely geometrical question respecting a property of cer- 

 tain ellipsoids. 



The author then enters into an analytical investigation, from 

 which he deduces the fundamental equation 



^~rTr^ = '^- rTv5 • <^-) 



the three axes of the ellipsoid being 



and A, B, C, constants, afterwards expressed by certain definite in- 

 tegrals. He then remarks that 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, in this case, the 

 centrifugal force will be represented in quantity and direction by a 

 line such that the resultant of this force and the whole attraction of 

 the ellipsoid upon a point in the surface will be perpendicular to the 

 surface. Lagrange had concluded that the equation (1), which re- 

 sults immediately from his investigations, admits of solution only in 

 spheroids of revolution, that is when A = A' and B = C ; but by ex- 

 pressing the functions A, B, C in elliptic integrals, M. Jacobi has 

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

 particular relation to one another. In order to ascertain the precise 

 limits within which this extension of the problem is possible, and to 

 determine the ellipsoid when the centrifugal force is given, the au- 

 thor has recourse to the equations of Lagrange, which contain all 

 the necessary conditions, and he deduces the equations 



/ = B-j^^, .-. . (2.) 



where / represents the intensity of the centrifugal force at the di- 

 stance equal to unity from the axis of rotation, and remarks that 

 these equations coincide with the equations of Lagrange. Substi- 

 tuting for A, B, C certain definite integrals given in the Mdcanique 

 Celeste, he deduces three equations expressing the value of g, the 

 ratio of the intensity of the centrifugal to that of the attractive force, 

 one of these being expressed in terms of the density and the other 

 two in the form of definite integrals; and then remarks that " these 

 equations comprehend all ellipsoids that are susceptible of equili- 

 brium on the supposition of a centrifugal force." 



He then applies these equations to the more simple case of the 

 spheroid of revolution, where X = x' = I, and determines the value 

 of I 



1= 2-5293, 



