858 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
"Hie first 2 being from iv to i, and the last only for ii and i. 
This is only a partial solution, and gives the result of the action of the moon on her 
own tides ; that for the sun on his tides may he obtained by symmetry. 
It is easy to see that for the joint effect 
di 
dt 
2r t ' 1 
9 (1— ^) 6 (1— v'T 
J 0 J 0 '[2P^(P 2 -<?*)F sin 2f+P(?(P 2 -Q 2 ) 3 G sin g] 
(300) 
From (299, 300) and (287-8) the complete solution may be collected. 
Then if these solutions be applied to the case where the earth is viscous and where 
the viscosity is small, it will be found after reduction as in previous cases that 
dn sin 4f 
r 2 (l -1 sin 3 7)(1 + 15T ? +±fV)+T ,a (l-£ sin^(l + 15V+ i f j y 2 ) 
-r 3 ^ cos i(l+27r)+273f)- r' 2 ^- cos 7(1 + 27t/+273?/ 2 ) 
~b n'g sin 2 7(1 fi- 3r)-\-3y)' -\-6r)' 2 ) 
(301) 
di sin 4f 
n- = —— 
dt 4a 
sm i cos i 
+1 +1V+W) 
— 2t 2 — sec 7(1 -h27p + 273p 2 ) —2t 3 ^- sec 7(1 + 27V+273V 3 ) 
11 12 / 
— tt / ( 1 -\-‘3r)-\-3r)' -\-6r) : -\-9rjr}' 
(302) 
These results give the tidal friction and rate of change of obliquity due both to the 
sun and moon ; rj is the ellipticity of the lunar orbit, and rj of the solar (or terrestrial) 
orbit. 
If rj and r{ lie put equal to zero they agree with the results obtained in the paper 
on “Precession. ’ 
§ 27. Verification of analysis, and effect of erectional tides. 
The analysis of this part of the paper has been long and complex, and therefore a 
verification is valuable. 
The moment of momentum of the orbital motion of the moon and earth round their 
common centre of inertia is proportional to the square root of the latus rectum of the 
orbit, according to the ordinary theory of elliptic motion. In the present notation this 
moment of momentum is equal to C£(l — y)/h Let us suppose the obliquity of the 
ecliptic to be zero. Then the whole moment of momentum of the system (supposing 
only one satellite to exist) is 
