34-36] The Rotational Problem 37 



I. FREELY ROTATING ELLIPSOIDS 



36. The necessary and sufficient condition that the standard ellipsoid 

 (51) shall be a figure of equilibrium for a homogeneous mass j)f jlensity p 

 freely rotating with angular velocity &> is that 



*) .............................. (62) 



shall be constant over the boundary, V- t being given by equation (57). Con- 

 sider the function 



+ |] + ^-l) ............ (63) 



where 6 is a constant, as yet undetermined. Operating with V 2 , we find 

 that this function will be a spherical harmonic, if 



- 47T/3 + 2o) 2 + 20>rrpabc + + - = 



\d- C / 



and this can be satisfied by assigning to 6 the value 



abc ( + ,- + - 



v + i) 



(64). 



Giving this value to 0, expression (63) becomes harmonic. The necessary 

 and sufficient condition that the standard ellipsoid (51) shall be a figure 

 of equilibrium is- that this function shall have a constant value over the 

 boundary. The function being harmonic, this is equivalent to the condition 

 that the function shall have a constant value throughout the interior of the 

 ellipsoid. We must accordingly have 



- irpabc ( 



+ 07rpabc 4- 1 + *- - 1 = cons. 



where is given by equation (64). Equating coefficients of # 2 , y 2 and 2 , this 

 equation is seen to be equivalent to the three separate equations 



........................... (65) ' 



Jo = .............................. (67). 



By addition of corresponding sides we again obtain equation (64) which 

 gives the value of 0. Thus the three equations (65) (67) contain within 



