1892.] Maclaurins Liquid Spheroid. 39 



where <^ is obviously the velocity potential of the irrotational 

 motion which would take place if the case were filled with liquid 

 at rest, and then set in rotation about the axis of z with angular 

 velocity H. 



The value of ^ is 



^hlhf.^' (2). 



also if h be the angular momentum, 



^'= p ijjx^'^''' -^y'^'+'^'^-y ^} '^'^'^y'^^ ^^^' 



and since by hypothesis ^ is independent of x, y and z, we obtain 



« = ^r-|/r + ip//*fciS (4), 



where / is the moment of inertia of the disturbed figure about the 

 axis of z. Since depends upon the disturbed motion, it must 

 necessarily be a linear function of small quantities of the first 

 order, whence (f>d(f)jdn is of the second order, and consequently the 

 surface integral may be taken over the original surface. 



The molecular rotation ^ must be determined from the con- 

 dition that the vorticity remains constant ; and since by hypothesis 

 the vortex lines remain parallel to z in the beginning of the 

 disturbed motion, and ^ is independent of x, y and z, we obtain 



^? = ^o?o=T(say) (5), 



where A is the area of the curve of intersection of the disturbed 

 surface with the plane {xy). 



If, therefore, the velocity potential <^ can be calculated, fl and 

 ^ can be eliminated from (4) by means of (8) and (5); and the 

 resulting value of ^ will be a homogeneous quadratic function of 

 A and T, whose coefficients are functions of the coordinates of the 

 disturbed surface. 



18. The reader who is acquainted with the methods by which 

 the steady motion of the ellipsoidal figures is investigated (see 

 Hydrodynamics, vol. ii. ch. xv.), and the method by which the 

 stability of such figures has been ascertained in the case of an 

 ellipsoidal displacement, will experience no difficulty in following 

 the preceding argument. It will, however, be noticed that the 

 method of § 17 depends upon certain conditions being fulfilled, 

 and when this is not the case a different course of procedure is 

 necessary. Although it is impossible to alter the vorticity of a 



