Mr. Ivory on the Attractions of homogeneous Ellipsoids, 357 



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 

 A A 



f = ^-T+T^' /=C-3-p^,. . . . (2.) 



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

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

 these equations coincide with the equations of Lagrange. Substi- 

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

 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 equilibrium 

 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' = /, and determines the value 

 oil 7=2-5293, 



and the corresponding maximum value of g = 0*3370, and re- 

 marks that, with respect to spheroids of revolution, it thus ap- 

 pears that an equilibrium is impossible when g, or its value in terms 

 of the density, is greater than 0*3370. In the extreme case, wheug 

 is equal to 0*3370, there is only one form of equilibrium, the axes of 

 the spheroid being 



A: and A: v/{l + (2*5293)2} or 2*7197 A; 

 but when g is less than 0*3370 there are two different forms of equi- 

 librium, the equatorial radius of the one being less, and of the other 

 greater than 2*7197 k, k being the semi-axis of rotation. 



The number of the forms of equilibrium in spheroids of revolution, 

 he remarks, is purely a mathematical deduction from the expression 

 of the ratio of the centrifugal to the attractive forces ; and as this has 

 been known since the time of Maclaurin, the discussion of it was all 

 that was wanted for perfecting this part of the theory. 



Returning to the general equations of the problem, the author 

 deduces the equations d<p 



d<5 



