464 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



On substituting this value for / in (20), neglecting squares of b and equating 

 corresponding harmonic terms, we obtain 



A '/ T / \ A,. SHI Kd lnc^\ 



<r = AOGT M./. (irf*) = -7=- -, ....... (22) 



- {-'/' J, /2 (*) I = *A /,>a-'M, /2 ( K a), . . . (23) 

 whence 



6 y = A.a- 1 ^ J ^M J '/.M ...... (24) 



* J/, (* ) 



By integration, the value of V m at a point on the sphere r = a is found to be 



v - - 



A _ +8/2 J B+ y, (<ra) Jy, (<ra) o 



" 



while the value of V, as given by equation (13), is 



V = ^a-'/'J 1 ,M+^a- 1/2 J + -,MS 1 ,+ cons. 



AC AC 



If we put 



V-V m = v = w + v,S 



we obtain, after some reduction, 



/ \ J, + ./.MJ./.M 



l 



In general, this gives the value of A n which determines the tide raised by a field of 



/r\* 

 potential v n (-j 8, proportional to S B . We notice that when n = 1, v n = or A B = oo 



\a/ 



independently of the values of /c, a. This merely expresses the obvious fact that 

 there can be no equilibrium at all so long as the fluid is acted on by a force 

 proportional to a harmonic of the first order. 



If it is possible for there to be a configuration of equilibrium when v n = 0, other 

 than that given by A B = 0, this configuration will of course determine a point of 

 bifurcation in the series of symmetrical configurations. The points of bifurcation are 

 accordingly given by v n = 0, or by 



J n -'/ g (KO) _ Jy, (KO) 

 *+'/,(*) J '/ 2 (*)' 



For brevity in printing, let us introduce the function u n defined by 



