1892.] Maclavriv's Liquid Spheroid. 81 



h^l2l to (12), which may be regarded indifferently as kinetic 

 energy or as potential energy due to centrifugal force ; whence if ^o 

 be the value of E for the original figure 



.=x.i./f..'.«-i//^.^'t;/^ (13). 



The second integral in (13) is equal to 



facr'dSdS' 



R 



Now a' is a function of the position of a point on the surface 

 of the original figure, and may, therefore, be regarded as a surface 

 distribution of matter ; whence if U be its potential 



(cr dS jj 



and y (14), 



dU, dUn . , 



-^ 5— = — 47ro- 



dn dn 



where U^, U^ are the values of JJ just outside and just inside the 

 surface. The integral may, therefore, be written 



p'jLudS; 



also since (/ - /q)^ is a quantity of the second order, we may write 

 I = Io in the denominator of the term in which this quantity 

 occurs, whence 



E = E, + l pjjga^dS - I p^ ljUadS+^'^^^-^^^ (15). 



The right-hand side of this equation represents the capacity of 

 the disturbed figure to do work on itself; and as the condition of 

 stability of the original figure requires that E^ should be a 

 minimum, the figure will be stable provided the sum of the last 

 three terms is positive, and unstable if it is negative. 



Comparing (15) with the result given by Poincar^ on p. 315, it 

 will be observed that he has omitted the last term of all. This 

 term, as will be presently shown, leads to the result that Maclaurin's 

 spheroid is stable for a spheroidal displacement. 



9. Poincare has applied (15), with the last term omitted, to 

 investigate the stability of a Jacobi's ellipsoid. The potential U 

 is expressed in a series of Lame's functions, and the value of a is 

 then found from (14), and the condition that the sum of the 

 second and third terms of (15) should be positive, leads to an 



