6 
SIR G. H. DARWIN: FURTHER CONSIDERATION OF THE 
It is shown on pp. 486, 487 of “ Harmonics” that 
<12 _ 
1 
3}{t, 4} 
<b 
4. l 2 + /3a— 
4. 2 2 +/3a —... 
to 
K> 
It. 
1 
1 
i/3 2 {i, 5} {i, 6} 
4.2 2 +{3a- 
i.3 2 + /3a—... 
%q e _ 
1 
8} 
A ' 
4 .3 2 + (3a— 
4.4' + /3 a — .. 
It may be remarked that the factor 2 occurs in each of these equations on the left, 
excepting in the first one ; also we are to take q 0 = 1. 
In the course of the successive approximations for the determination of /3a, each 
of these fractions is naturally evaluated. Therefore it is only necessary to extract 
certain numerical values already found in the course of solving the equation for /3a. 
As a verification, which shows whether the equation has been correctly solved, we 
have 
<b = WH f}{D 2} . 
q 2 /3a 
It is now obvious that we are able to find all the q’s in terms of q 0 , which is unity. 
1 — k 2 
We then multiply each q by its appropriate power of (3 or -that is to say, we 
form /3 r q 2r for r = 1, 2, ..., and introduce the results into the formula for 
A closely analogous method enables us to find all the other types of function for an 
ellipsoid of known ellipticities, but, except for certain harmonics of the fourth order, 
it is not possible to obtain rigorous analytical solutions. Approximate analytical 
forms are given in “ Harmonics,” and the approximation may be carried further if 
desired. 
The following tables give the coefficients in the several functions for the. critical 
Jacobian ellipsoid with which we are dealing 
