752 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
This is only a partial solution, since it only refers to the action of the moon on her 
own tides. 
If the second satellite, say the sun, he introduced, the action of the sun on the solar 
tides may be written down by symmetry, and the elements of the solar (or terrestrial) 
orbit may be indicated by the same symbols as before, but with accents. 
From (85) and (81) the complete solution may be collected. 
In the case of viscosity, and where the viscosity is small, it will be found that the 
solution becomes 
■Jt = \ -i sin3 { ) (t 2 +7 /£ )—1(1. -f sin 3 i) (r 3 sin 3 J+r' 3 sin 3 /) 
— t 3 — cos i cos j — t' 2 — cos i cos/+/rr / sin 3 7(1 — f sin 3 / (1 — | sin 3 /) j . (86) 
TL Tt 
If j and / be put equal to zero and SI' /n neglected, this result will be found to agree 
with that given in the paper on c ‘ Precession,” § 17, (83), 
We will next consider the change of obliquity. 
The combined effect has already been determined in (81), but the separate effects of 
the two bodies remain to be found. The terms of different speeds must now be taken 
one by one. 
Slow semi-diurnal term. 
di . Q dW l | 1 dW 1 
H dt~ g — P ~de r ' 2PQ IN 7 
We had before 
dhVj 1 clW j 
k dt ‘ g p de' 2\pq dN 
Now Wj is symmetrical with regard to P and p, Q and q, and so are its differentials 
with regard to e and N'. The solution may be written down by symmetry with the 
“ slow semi-diurnal ” of § 6, by writing P for p and Q for q and vice versa. 
Let 
Fi=i{P 6 p 8 -4P 4 (P 3 -3^>Y- 1 8P 2 ^ 2 (P 2 -Q 3 )pY-4Q 4 (3P 2 -Q 2 )pY-W} ( 8 ^) 
and 
wf H- — = 2FiFi sin 2f sin i . (88) 
dt g 
Sidereal semi-diurnal term. 
di t~ 1 
d,W„ . dWP 
