756 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
The complete solution of the problem may be collected from the equations (101) 
and (81). 
In the case of the viscosity of the earth, and when the viscosity is small, we easily 
find the complete solution to be 
3 * o 2/2 rt 
•f smq )-—r" sec i cos j 
f sinV)(l—I sin 2 /) | . . (104) 
di sin 41 7 . . • f q ■> 9 ‘\ i /o/-i 
01 - 7 = - \ sm ^ cos il r~( 1 — f sur/fi-T \l — 
dt 
2/2' /g . 
— t “ sec l COS^ —TT (1 — 
This result agrees with that given in (83) of “ Precession,” when the squares of 
j and / are neglected, and when D!fn is also neglected. 
The preceding method of finding the tidal friction and change of obliquity is no 
doubt somewhat artificial, but as the principal object of the present paper is to discuss 
the secular changes in the elements of the satellite’s orbit, it did not seem worth while 
to develop the disturbing function in such a form as would make it applicable both to 
the satellite and the planet; it seemed preferable to develop it for the satellite and 
then to adapt it for the case of the perturbation of the planet. 
In long analytical investigations it is difficult to avoid mistakes; it may therefore 
give the reader confidence in the correctness of the results and process if I state that 
I have worked out the preceding values of di/dt and dn/dt independently, by means of 
the determination of the disturbing couples U, /hi, ■$,. That investigation separated 
itself from the present one at the point where the products of the X'-Y-Z' functions 
and functions are formed, for products of the form Y'Z' X ^ had there to be 
found. From this early stage the two processes are quite independent, and the identity 
of the results is confirmatory of both. Moreover, the investigation here presented 
reposes on the values found for dj/dt and d^/dt, hence the correctness of the result of 
the first problem here treated was also confirmed. 
Ill 
THE PROPER PLANES OP THE SATELLITE, AND OP THE PLANET, AND THEIR 
SECULAR CHANGES. 
§ 13. On the motion of a satellite moving about a rigid oblate spheroidal planet, and 
perturbed by another satellite. 
The present problem is to determine the joint effects of the perturbing influence of 
the sun, and of the earth’s oblateness upon the motion of the moon’s nodes, and upon 
the inclination of the orbit to the ecliptic ; and also to determine the effects on the 
