674 



Dr. J. R. Wilton 



Figures of 



we have, on substituting for vr 2 in equation (1), and equating 

 the coefficients of the various zonal harmonics to zero, the 

 system of equations 



I 





dx = C + I'ao? 



rPi d# = ^a x , 



P 2 log r dx = vu 2 + {i —v)=vu 2 + ^ — 



(A) 



and, if n > 2, 



j: 



p «z^+(^-2)^=0. 



The first of these equations merely determines the value 

 of the constant in the equation Y + ^co 2 R 2 = const., and may 

 be discarded. The others determine the form of the surface 

 of equilibrium. 



2. We know beforehand that a particular solution of 

 equations (A) will be the series of Maclaurin spheroids, for 

 which a 1 = a s = a 4L = cc 5 = ... =0, and it is easy to verify that 

 these values do, in fact, satisfy the equations. The value of a 2 is 

 not, however, arbitrary, but is connected with that of a) 2 /27rp 

 by the equation 



j: 



(3) 



It 



P 2 logrdx = va 2 +-~ — .. . . 



6 ZTTp 



11 be convenient to put 



a 2 =(-2P + 3a 2 )/(3 + FJ, and 



«„ = 3a»/(3 + A 2 ), 0=^2) 



so that equation (2) takes the form 



r\l + k 2 x 2 ) = a + a 1 rP 1 + a 2 r 2 F 2 + ... a n r»P n + ... 



where in the case of Maclaurin's spheroid a n = 0, if n^O. 

 Equation (3) then becomes (since for the spheroid a 2 = 0) 



(2') 



P 2 log r dx 



1 



2Pv 



or, if K = v/(1 +£#»), 

 3 2^ .3*^ 



r 



3 2irp 3 + P' 



P 2 log(l + £V 9 )^ 



1 + k 2 



(^-tan- 1 ^), 



