THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 47 



For the solution fa = K of degree n = 0, the limit at infinity of the contribution 

 to the potential is 



f" Trpabc-, 



K I -- - - d\ 



r 



+ ..., 



showing that the distortion involves a change of mass SM. = ^TrpabcK, accompanied 

 of course with a change in the inertia terms. 



For the solution (j> b -^ of order n = 1 , 



a 



,-rr f* Trpabc 7 . 7 a?x i ^mx 



00 = X Jx A A = ~ 5 ^ T 5 " ' ~ ^ ~^' 



so that this distortion represents a motion of the centre of gravity by an amount 



XT - _l./y 2 



O Ki/0 2 ^ 



Solutions of degree n = 2 will clearly involve changes in the moments and products 

 of inertia. The limiting potentials are found to be as follows : 



(i) A- J|, 



(ii) 4 = ^j , 



Cv / v / 



The first solution does not involve a change in mass, whilst the second does ; both 

 distortions affect the inertia. 



For the solutions of degree n = 3, the limiting values are as follows : 



This distortion changes neither mass, centre of gravity, nor inertia. 



- - w abc 



^ 



i ^l 



/yi'J /y /y> 



(iii) ^ = - , S V x = - frp abc f - 7 - | TT P abc -^ 



\M I C& I 



These two latter distortions move the centre of gravity, but do not affect the 

 mass or inertia. 



It is clear, without detailed examination, that the distortions represented by 

 solutions of degree n = 4 cannot affect the centre of gravity, but may affect the 

 mass and inertia. 



