1892.] Maclaurins Liquid Spheroid. 37 



Substituting the values of q^, q^, p^, the condition becomes 

 37 + 97' - (- 1 + 67' + 97*) cot-' 7 > 0, 

 or 9e (1 - e'f (3 - 2e') - (27 - 36e' + 8e') sin"' e > . . .(28). 



This condition is the same as the condition of stability of a 

 Maclaurin's spheroid composed of frictionless liquid, when the 

 displacement is spheroidal (see Hydrodynamics, vol. ii. p. 124), 

 and consequently the spheroid is stable for this kind of displace- 

 ment. 



Stability of Figures compiosed of Frictionless Liquid. 



14. The investigation of the stability of a rotating mass of 

 frictionless liquid by means of the energy method depends upon 

 totally different principles. It must be borne in mind that in 

 steady motion it is not essential that the liquid should rotate as a 

 rigid body, and one figure is known (the irrotational ellipsoid) 

 in which the motion does not possess molecular rotation. On the 

 other hand, if the liquid is viscous, and is acted upon by no forces 

 except those due to its own attraction, steady motion cannot exist 

 unless the liquid rotates as a rigid body ; for if any different 

 motion existed at any particular instant, it would gradually be 

 extinguished by viscosity, and the final state would necessarily be 

 a rigid body motion. 



15. Whenever the momenta of a conservative dynamical 

 system, or any quantities in the nature of momenta, are con- 

 stant throughout the motion, the steady motion and stability 

 may be investigated in the following manner*. A constant 

 momentum always involves an ignored coordinate, and if the 

 velocities corresponding to the ignored coordinates be eliminated 

 from the Lagrangian expression for the kinetic energy, the result 

 will be of the form X + £, where 2^ is a homogeneous quadratic 

 function of velocities 6 which are the time variations of co- 

 ordinates which appear in the expression for the total energy 

 of the system, and ^ is a similar function of the constant momenta. 

 Under these circumstances, it follows that if V be the potential 

 energy of the system measured from a configuration of stable equi- 

 librium, the steady motion will be obtained by assigning constant 

 values to the coordinates 6, and making ^ + V stationary ; and 

 the steady motion will be stable, provided £ -F F is a minimum. 

 The function ^ is accordingly the kinetic energy of the most 

 general possible motion which the system could assume if the co- 

 ordinates 6 were constrained to remain invariable, and the function 

 £■ -I- F is the total energy of that motion. 



* Froc. Cainb. Phil. Soc, vol. vii. p. 361. 



