466 ME. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



where P 2 is the second zonal harmonic about the axis of rotation as 6 = 0. Assuming 

 the free surface to be 



r = a + bP 2 , ...... ..... (30) 



the equations analogous to (22), (23), and (24) are found to be 



or-I^Aoa-'/'J^M, ..'.-. ...... (31) 



A 2 a-' /8 J s/2 (fc) = K^la-^S^Ko), ........ (32) 



** . (33) 



l W \T / \ 



1 - M) 



- 



Let v be given by 



v = V+ty> w '-V m = V-V,, I + ^V(1-P 2 ), 

 then, instead of equation (25), we have 



(34) 



in which constant terms are omitted, and the value is taken on the sphere r = a. 

 For a configuration of equilibrium under no external field of force we must have 

 V = V m , and therefore v in equation (34) equal to -^wVP 2 . Neglecting squares 



of w 2 , and therefore omitting the factor 1 - ^ in the denominator, the equation 



ZTTIT 



becomes 



(35) 



giving A 2 in terms of w 2 , when w 2 is small. It will be readily verified that this 

 equation is identical with that obtained by THOMSON and TAIT ('Nat. Phil.,' 824, 

 equation (14)). 



13. We next examine the solution 



w 2 



X = /'-^ = A r-^J 1/2 (^-) + A 2 r-^J s/2 (^)P 2 + A B r- 1 /-J B+1 ,(/cr)S B , . . (36) 



which is appropriate to a mass of fluid having a rotation w given by equation (35), 

 and acted on by a field of force of potential v n $ a . By analysis exactly similar to 

 that just given, we obtain at r = a 



