CHAPTER VII 
NON-SPHERICAL MASSES—DYNAMICAL PRINCIPLES 
Configurations of Equilibrium. 
168. The last four chapters have been devoted to a discussion of the 
commonest object in the sky—the simple star. Its mechanism and build were 
discussed on the supposition that it was of spherical shape. 
We turn now to the discussion of non-spherical masses. A star which was 
initially spherical may assume other shapes as the result of being set into 
rotation, of coming within the field of gravitational force arising from some 
other body, or in other ways. Our main problem is to calculate the shapes 
assumed by astronomical bodies under such circumstances, the ultimate aim 
being, of course, to compare the calculated shapes with the observed shapes of 
astronomical objects. The discussion will no longer be limited to stars. The 
present chapter will contain an exposition of dynamical principles which we 
shall subsequently apply to nebulae, stars, the sun, the earth and the planets, 
although the principles are so general as to apply to all dynamical systems and 
not merely to astronomical bodies. 
169. Let the configuration of a dynamical system at any instant be fixed 
by a number of co-ordinates 
o„o 2 ,o 3 ,...o n (ie> 9 i), 
while its motion is specified in terms of the corresponding velocities 
On (169-2). 
The potential energy W of the system is a function of the co-ordinates of 
position only, say 
W=f(0 1 ,0 2 ,...0 n ) (169-3), 
while the kinetic energy T is a function both of the co-ordinates of position 
and of the velocities, say 
T=F($„e t ,... e„, à„ i„... 4) (i69 ; 4), 
this function being of the second degree in the velocities 0 lt 0 2 , ... 0 n . 
The motion of the system will be determined by the usual Lagrangian 
equations 
where F u F 2 , ... F n are the “generalised forces” applied from outside. 
In a great number of cosmogonical problems we are concerned with 
astronomical masses which are either in a state of equilibrium or whose 
dt 
(dT\ dT dW „ . 
(k)-a* r™. +F ‘ (s=1 ’ 2 -" n) 
.(169-5)