480 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Now to obtain p,—, we liave to add the last four expressions together, and we 
observe that the last two cut one another out, so that the expression for -j is inde¬ 
pendent of the solar tides; also the terms in sin 2e, sin e cut one another out in the 
... . . dP 
sum of the first two expressions, and hence it follows that is independent of the 
sidereal semi-diurnal and diurnal terms. 
Thus we have 
ji-j f = ~[_Ei r p *sin 2e, ■-Epf sin 2e 3 + 4T7 1 y> f V J rsin e\ — 4E 2 pr(f sin e' 2 — QE'p^cf sin 2e"j (57) 
This equation will be referred to hereafter as that of tidal reaction.* From its form 
we see that the tides of speeds 2(n+/2), n-\- 2/2, and 2/2 tend to make the moon 
approach the earth, whilst the other tides tend to make it recede. 
Then if, as in previous cases, we put Ep=E 2 —E\ E\— E ' 2 = E'; e 1 =e 3 =e; e / 1 =e' 2 =e' 
(which is justifiable so long as the moon’s orbital motion is slow compared with that of 
the earth’s rotation), we have, after noticing that 
p s — q 8 = (p 2 —q 2 ) (p Aj r ( f) — cos &'(1 — \ sin 2 i) 
4p & q 2 — 4p 2 q 6 = 4p 2 q 2 (p 2 —q 2 ) = sin 2 i cos i 
6^dq 4 =-| sin 4, i 
Pp = -—[cos i(l — ^ sin 2 i)E sin 2e+ sin 2 i cos iE' sin e—-§ sin 4 iE" sin 2e"J . (58) 
at Q'TZ'o 
Now if the present values of n, fl, i be substituted in this equation (58) (i.e., with 
the present day, month, and obliquity), and if the tropical year be the unit of time, it 
will be found that 
10 10 ^=;^(24'27i£ sin 2e-|-4T8i7 sin e — *27 lE" sin 2e") 
at p* 
£ 13 enters into this equation because r varies as fl 2 and therefore as £ -6 . 
But we may here put C— I, because at present we only want the instantaneous rate 
of increase of PL. 
Now - ( -~ when J2 = /2 0 ; hence multiplying the equation by 
3/2 0 we have at the present time 
6 . 
—• 10 10 -^=6115i? sin 2e-b 1053.E 7 sin e —6S'28E" sin 2e" . . . (59) 
Cl L 
m radians per annum. 
* In a future paper on the perturbations of a satellite revolving about a viscous primary, I shall obtain 
this equation by the method of the disturbing function. 
