356 Mr Basset, On the Steady Motion [May 30, 



whence £= ( ^^ + ^, 



and V=(M-M')gy. 



The equation of steady motion is 



4<a+F)-o. 



which leads at once to 



pu 2 - /cpu coth a + /c 2 /)/4ttc + (if - M ') # = 0, 



where P~ ~ h dRjdy, u = ob, 



which is equation (10), § 213. 



The stability of the various cases which arise, can be found 

 from the condition that 



Jp (ffi+F) 



should be positive, but the calculation would be somewhat trouble- 

 some. If however the radius of the cylinder were small in com- 

 parison with its distance from the wall, approximate results might 

 be obtained in a fairly simple form. 



9. In Hydrodynamical problems, which involve molecular 

 rotation, certain quantities occur, which although not properly 

 speaking momenta, are quantities in the nature of momenta ; and 

 to this class of quantities vorticity belongs. Let a surface 8 

 be drawn in a liquid, cutting each vortex line once only ; let co 

 be the resultant molecular rotation, e the angle between the 

 direction of &> and the normal to dS drawn outwards. Then the 



integral 



r f 



o) cos edS 



//■ 



is constant throughout the motion, and this statement expresses 

 the fact that the vorticity of the mass of liquid is constant. The 

 dimensions of this quantity, when multiplied by p are \ML 2 T~ 1 \ 

 and these are the dimensions of an angular momentum. We 

 may therefore regard the constant quantity 



p \\ a cos edS, 



as a generalized component of momentum. 



When a liquid ellipsoid is rotating about a principal axis 

 (say c), the value of this integral is 



irpab^= 7rpR 3 ^fc, 



