8 
SIR G. H. DARWIN: FURTHER CONSIDERATION OF THE 
Coefficients of Terms in (g) when expressed in Sines and Cosines. 
i. s. 
/o- 
u 
U 
u 
fs■ 
fw • 
6 0 
8 0 
10 0 
1•00000 
1-00000 
1-00000 
9-45814 
19-29797 
33-29978 
7-63713 
40-01416 
131-74546 
0-44035 
14-03323 
116-85134 
0-45235 
22-76398 
0-46728 
§ 2. The Rigorous Expression for the Harmonics of the Second Kind. 
The integral <H/ denotes p/(i> 0 ) ©/(v 0 ), and 13/ denotes P/(d>)- Thus 
clv 0 
'll/ is, in fact, the harmonic function of the second kind. $3/ is clearly determinable 
from ^/. 
In the original paper iH/ was found by quadrature, and this defect in my procedure 
is referred to by M. Liapounoff as a cause possibly contributory to the discrepancy 
between our results. Quadrature was not, perhaps, a very satisfactory method, and 
the defect will now be made good by finding these integrals in terms of the F and E 
elliptic integrals. It will appear that my former results were sufficiently near to the 
truth for practical purposes. 
The functions |3/ (v) or P/(Q are of eight types, determined by the oddness or 
evenness of i and s, and the association with a cosine or sine function of <£. In 
“ Harmonics” the types are indicated by combinations in groups of three of the four 
letters E, O, C, S—denoting Even, Odd, Cosine, Sine ; for example, OES means i odd, 
s even, associated with a sine function. 
All the roots of the equation ^/(Q or P/(Q = 0 are real, and when the form of the 
function has been determined by the method of § 1 the equation may be solved. 
Hence these functions are expressible as the products of a number of factors ; and it 
is to be noted that it is not necessary to adopt the same definition as in “ Harmonics,” 
because the function may be multiplied by any constant factor, without affecting the 
result. 
For brevity, let 
n (o 2 -g, 2 ) = (FF-q^) (FF-qf )... (kV-^V 3 ), 
where F is (J — /3)/( 1 +/3) of “Harmonics.” The parameters, i< and y, as elsewhere, 
define the form of the ellipsoid. 
An alternative notation will be needed, in which we write 
1 
F sin 2 xfj' 
v " = 
Ay = 1 — q x 2 sin 2 xJj, A 2 = 1 — F sin 2 xfj- 
