FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
23 
The solution of these equations is found to be 
H= -12*6275, il = -0-0307056, N = -0'0044636, 
l = — 0*0116194, nt = 0M2602, it = 1*15365. 
These values have been checked by insertion, not only in the six equations from 
which they were directly derived, but also in the remaining equations (73) and (74). 
We now have all the material necessary for the evaluation of Itj, lt 2 and Ht 3 , 
which, it will be remembered, are the right-hand members of equations (77)—(79) 
respectively. These have been computed independently, and I found 
% = -0-0016803 + 0-228894n", 
% = +0-0026780 —0-30111072", 
%= +0-0 0 2 3 7 3 9 - 0-3 8 5 6 4 472". 
These values ought to satisfy (cf. § 16) 
33^!+3a2~83&3 — 0 , 
in place of which I find 
3% + % + % = 0-0 0 0 011 -0-0 0 0 0 772", 
but the error is no greater than might reasonably be expected in view of the very 
large number of operations in each computation.* 
The coefficients in equation (90) are found to be 
—M 2-Tih = 0-0058753, = 0*00024949, 
« 4 6 4 a b 
so that the equation itself becomes 
5-8753 (-0-0016803 + 0-22889472") = 0’24949 (0-0026780-0-30111022"), 
and the solution is found to be 
n" = 0-007423. 
Completion of Second Order Solution. 
18. On substituting the value just obtained for n" into the values for p, q , r, s, 
found in the previous paper (§ 34), I find, 
p = 3-124954, 
q = -0-103164, 
r = -0-015236, 
‘ .$=-0-256962, 
thus completing the figure to the second-order terms. 
* Each of the quantities 11 b 1+, 11.3 has been computed by expressing it as a sum of integrals of the 
type tabulated on p. 21. The first term in 1+, namely 13iK« 2 a _4 J AAA , may be thought of as a typical 
term. Each of the quantities !+, U 2 , M 3 consisted of 326 such terms, so that 3i\i + HU + M 3 is a sum of 978 
such terms, each of which, it must be remembered, is evaluated by a fairly lengthy series of computations. 
