196 
SIR G. H. DARWIN ON THE 
fi became infinite when X = 0. But this is not so, because when €li S (r/k) is developed 
the first term of the series is one in (k/r ) l+1 ; now Jc 3 = — a 3 sin 3 fi sec fi sec y, and 
1 + X 
therefore the formula for f { s involves the factor ( —— ) . - or ( 
\1 + X/ X Vl+X/ 1 —f— X 
We see then that fi s vanishes both when X = 0 and X = oo. 
This factor is a maximum when X = Therefore we should expect, cceteris 
paribus, the third harmonics to be most important when X = the fourth when 
X = f, the fifth when X = 1, and the higher harmonics when X is greater than unity. 
I his prevision is partially fulfilled by the numerical results given below, but it was 
not to be expected that it should be exactly so, because the other conditions are not 
exactly the same in the solutions for various values of X. 
The formula shows, as stated above, that f s is of order r~ i ~ 1 . The series in the 
denominator affects the result but slightly and might be omitted, except, perhaps, in 
the case of the third zonal harmonic. For all harmonics other than cosine-harmonics 
of even rank f* is zero. 
It is now possible to eliminate f/ from (vv) by substituting for it its value. These 
terms in (vv) become, in this way, equal to 
(f 7rpa 3 ) 2 
X 
(1+X) 2 
2i+ 1 
2kX 
Ki \%i— A d—series] 
• (26). 
When i = 3, this term is of order r~ 8 , and is negligible; hence we need no longer 
pay any attention to the inequalities on the ellipsoid. However, the formula (25) is 
important as rendering it possible to evaluate the inequalities. 
Since for all inequalities, excepting cosine-harmonics of even rank, f s only occurs 
in the energy function as a square, it is in these cases a principal co-ordinate, and 
W-Ai 1 — series ) is a coefficient of stability. 
But the like is not true for the cosine-harmonics of even rank, because, when we 
consider, for example, the harmonics of the third order, we see that 0 2 V/clf 3 s dr is of 
the fifth order and d 2 V/df 3 s 0a, 0H /df 3 s d ft (5 — 0, 2) are of the fourth order. 
It is clear that the inequalities on the ellipsoid E are determinable by symmetrical 
formulae. 
We must now turn to the equations of equilibrium for the parameters a and fj. 
Since differentiation with respect to these parameters is effected most conveniently 
by means of the formulae (23), the portion of V called ( eE) 2 should be written in the 
form (15). After effecting the differentiations it is, however, best to revert to the 
notation involving Jc, k, y, K^, K, r; but as an exception to the general rule as to 
notation, it is most convenient to retain the differentials of P(1 + 1 /k 2 ) and of b 2 +c 2 in 
the forms involving a, b, c. As the algebraic processes involved are rather long, I 
simply give the results, as follows :— 
