1892.] Maclaurin's Liquid Spheroid. 25 



But although the presence of the term Ih^/I, which may 

 be regarded as the potential energ}' due to centrifugal force, 

 increases the total energy of the system, it diminishes its capacity 

 to do work upon itself. For equation (3) may be written 



E=W.-{W-WII) (4), 



and since W^ is a constant, the capacity to do work depends upon 

 the last term of (4). Also if (4) be written in the form 



E=W, + {WII-W) (5), 



it follows that the capacity to do work will be least, when 



is a minimum. When this is the case, the system will be in stable 

 equilibrium ; for if it be disturbed in any manner, its capacity to 

 do work will be increased, and the system will therefore tend to 

 return to its configuration of relative equilibrium, and will perform 

 small oscillations about it. 



When the figure is passing through its configuration of relative 

 equilibrium, the portion of the kinetic energy which depends upon 

 the oscillations will be a maximum ; but if the liquid is viscous, 

 this portion will gradually be transformed into heat. Hence if 

 E + 8E be the energy immediately after disturbance, 8E must 

 gradually diminish and ultimately vanish ; and when 8E has been 

 entirely converted into heat, the figure will be restored to its 

 original state before disturbance. We have, therefore, the follow- 

 ing rule for determining the steady motion and stability of a mass 

 of viscous liquid which is rotating as a I'igid body in relative equi- 

 librium. Let the figure be displaced into a slightly different 

 form, and let Tf o — 1^ be the work which must be done against 

 gravitation in order to displace it from the spherical into its dis- 

 turbed form ; also let ^i /I be the kinetic energy of a mass of 

 liquid of the same form as the disturbed figure, which is supposed 

 to be rotating as a rigid body with unchanged angular momentum : 

 then the condition of steady motion is that ^h^/I — W should be 

 stationary, and the condition of stability is that this quantity 

 should be a minimum. 



3. We are now in a position to give a formal proof of the 

 proposition that Maclaurin's spheroid, when composed of viscous 

 liquid, is unstable for an ellipsoidal displacement if the excen- 

 tricity exceeds •8127. 



To do this, we must suppose that the liquid is rotating as a 

 rigid body about its least axis, and that the free surface is an 

 ellipsoid which differs very slightly from the spheroidal form, and 

 we have to find the conditions that F should be a minimum, where 



V=^h'/I- W. 



