810 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
In these equations we have, recapitulating the notation 
fen . ft 
m= 
Also 
— X= , C=i 
71 Cl 
71 " 
L + 'A— — a — /I 
k , — Ko= — \/ (a — /3)' : fl-4ab 
( 252 ) 
(253) 
Lastly we have by (186) 
i+ ; fpi-f)+A(I-)(i-P) 
(254) 
which gives parallel values of n and £ 
These equations will be solved by quadratures for the case of the moon and earth in 
Part IV. 
If t' r be so small as to he negligeable, and t'/2\Ct small compared with unity, then 
the equations (250) admit of reduction to a simple form. 
With this hypothesis it is easy to find approximate values of k x and k. 2 , and then by 
some easy, but rather tedious analysis, it may be shown that (250) reduce to the 
following;—- 
kn~ log tan |J = 
m + l r , t' 1 l + llm~l 
m 
t 2 X 0 (l + m) 2 
7 d it T 1 1 + llm 
fen— r log' tan if =— —— - 0 
cl% 2 r 2Xc (l + m) 2 
y 
(255) 
J 
These equations would give the secular changes of J and I, when the solar influence 
is very small compared with that of the moon. Of course if G be replaced by g, 
they are applicable to the case of small viscosity. 
It is remarkable that the changes of I are independent of the viscosity; they depend 
in fact solely on the secular change in the permanent ellipticity of the earth. 
IV. 
INTEGRATION OF THE DIFFERENTIAL EQUATIONS FOR CHANGES IN THE 
INCLINATION OF THE ORBIT AND THE OBLIQUITY OF THE ECLIPTIC. 
§ 18. Integration in the case of small viscosity, where the nodes revolve uniformly. 
It is not, even at the present time, rigorously true that the nodes of the lunar orbit 
revolve uniformly on the ecliptic and that the inclination of the orbit is constant; but 
it is very nearly true, and the integration may be carried backwards in time for a long 
way without an important departure from accuracy. 
