179, 180] GRAVITATING ROTATING FLUID. 469 



Inserting in the integral equation 11) (see footnote p. 466), 



32) ~ + i{ Lx * + W + N ^ - 7 (f + * 2 ) } = c t., 

 the surfaces of equal pressure are similar to the ellipsoid 



33) L<t 



from which we obtain 



o 4 N w 2 Iffe 2 - a 2 _ Nc*-La* _ Mb* - Nc* 



y & 2 c 2 6 2 -c 2 



Equating the first and third values of - 



35) (& 2 - c 

 or otherwise 



36) a 2 (c 2 -b 2 )L = b 2 c* (M- N). 



Since L, M and JV are transcendental functions of a, b, c, this is a 

 transcendental equation for the ratios of the axes. Since M is the 

 same function of b that N is of c obviously the equation is satisfied 

 if b = c and M = N, giving an ellipsoid of revolution as a possible form 

 of equilibrium. This is the celebrated solution given by Maclaurin in 1738. 

 If we put 



the formulae 60), 62), 160 give 



37) = 4^^ (A -tan- 1 A), 



M = N = 



Introducing these values the first value of in equation 34) becomes 



By the development of tan" 1 1 in a series we find 

 39) 



1 



= for I = and I = <x>, 

 and ^0) = for I = 2.5293, 



for which value 



