38 Mr Basset, On the Stability of [Nov. 28, 



16. When a mass of frictionless liquid is rotating in any 

 manner in steady motion, the fact of the motion being steady 

 requires that the form of the free surface should remain invariable, 

 and that it should rotate like a rigid body, although it is not 

 necessary for the particles composing the film of liquid which 

 forms the free surface to move in this manner. The kinetic energy 

 of the steady motion will accordingly be a homogeneous quadratic 

 function of constant quantities which specify the fact that the 

 generalized momenta and vorticity of the liquid are constant. 

 This is the function £. Now let the liquid be disturbed in any 

 manner, subject to the condition that the generalized momenta 

 remain unchanged; and in contemplating conceivable disturb- 

 ances, it must be recollected that although it is usually possible to 

 apply a disturbance which will alter one or more of the generalized 

 momenta, it is impossible to alter the vorticity. The new form of 

 the free surface will depend upon certain parameters 6 which are 

 coordinates, and which have certain definite values in steady 

 motion ; accordingly the kinetic energy of the disturbed motion 

 will be expressible in the form % + iJ, where S^ is a homogeneous 

 quadratic function of the velocities 6. The form of the function ^ 

 in the disturbed motion will not necessarily be the same as in 

 steady motion, since it frequently happens that in steady motion 

 some of the coordinates 6 are zero or are equal to one another. 



17. The function V, which is the potential energy due to 

 gravitation, can be calculated by Poincares method ; but the 

 calculation of ^ presents various difficulties. When the liquid is 

 rotating about a fixed axis, it sometimes happens that in the 

 beginning of the disturbed motion the vortex lines remain parallel 

 to that axis, and that the new value of the molecular rotation is 

 independent of the position of particular particles of liquid. 

 Whenever this is the case, the disturbed motion depends upon a 

 velocity potential, and can be determined by means of Prof 

 Greenhill's method. Let the liquid be enclosed in a case whose 

 form is the same as that of the free surface of the disturbed figure 

 at any particular epoch, and let the liquid be frozen and set in 

 rotation with angular velocity ^ about the fixed axis of rotation ; 

 let the liquid now be melted and an additional angular velocity fi 

 be impressed on the case, then ^ is the kinetic energy of the 

 resulting motion. 



The subsequent calculation is as follows. Let the motion be 

 referred to axes x and y fixed in the case, and which are therefore 

 rotating about the fixed axis of z with angular velocity w, where 

 ft> = H + f ; then if u, v, w be the velocities of the liquid referred 

 to this set of axes, 



dd> ^ d(b ^ d6 .,, 



