﻿in Small Orbits* 515 



and April 1861. For a more complete and accurate investiga- 

 tion some of the formulae which I then employed will be found 

 necessary ; and these I will briefly deduce. While retaining the 

 same notation as in the preceding case, I shall put P, Q, and R 

 for the force of the satellite's attraction at the extremities of the 

 axes, the major axis ranging with the centre of the primary. 

 That the figure of equilibrium is an ellipsoid will be rendered 

 evident by a comparison of the results to which this hypothesis 

 leads. Regarding the body as an ellipsoid, its attraction at the 

 point of the surface represented by x, y, and z will, by a well- 

 known theorem, have the following components in directions 

 parallel to each axis : 



Vx Qy R* 



X' B - ' "C" (15) 



From formula (2) the components of the primary's disturbance 

 at the same point will be 



Mk*x MAfy Mff* 



^~ J)3 * J)3 ' JJ3 > 



and the components of centrifugal force will be, according to 

 formula (3), 



D 3 ' J) 3 ' 



Calling the sum of the forces, in the direction of each axis, X, 

 Y, and Z, 



v Vx SMh 9 x v Qy A „ - B* t Mk*Z : 



In the equation of equilibrium for a fluid mass, 



Xdx + Ydy + Zdz=0; 



substituting the above values for X, Y, and Z, and integrating, 

 there results 



This being the equation of the surface of an ellipsoid, the 

 result justifies the course which has been pursued in assuming 

 this figure as the one to which a homogeneous fluid satellite 

 conforms. When z and y are both equal to nothing, x attains 

 its maximum value and becomes equal to A, and the last equa- 

 tion is reduced to 



PA_^A. 2 = s (18) 



The maximum value of y and of z being likewise determined by 



