468 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Tims 
It,,,,," 4 tt, __^ c — Xfgz —d(?/' — z~) — exy-\- fzx. 
Comparing with (14), when X is put equal to zero, we have 
% L '-~ = -iflvP'+i'P -HA+HA 
This quantity may be evaluated at once by reference to (15), (16), and (17), for it is 
clear that % /nm/ is what becomes when E x —E<y=. 0, E\ — E\ x — 0, and when 2 tt, 
replaces r 3 . 
If, therefore, we put^p= F mm sin n -f- G mm/ cos n, and remark that 
= 2 M (p 3 - S *)(/+2*-6p2 3 )=i > <?(l-2« 2 ), 2 pq(^-ff=P a Q, 
we have by selecting the terms in E, E' out of (15) and (16), 
F mr// -T- —= —| EPQ ?J cos 2e—E'PQ(l — 2Q' 2 ) cos e' 
> 
\EPQ? sin 2e+E'P B Q sin c 
S j 
(33) 
It may be shown in a precisely similar way by selecting terms out of (21) that 
— =±EQ 4 ' sin 2e-\-E'P~Q' 2 sin e 
(34) 
It is worthy of notice that (33) and (34) would be exactly the same, even if we did 
not put E x —E % —E\ E\—E\=E'\ e 1 =e 2 =e; e , 1 =e / 3 =e / , because these new terms 
depend entirely on the sidereal semi-diurnal and diurnal tides. The new expressions 
which ought rigorously to give the heights and lagging of the solar semi-diurnal and 
diurnal tides would only occur in 
In the two following sections the results are collected with respect to the rate of 
change of obliquity and with respect to the tidal friction. 
§ 9. The rate of change of obliquity due to both sun and moon. 
cLt 
The suffixes m 3 , m~, mm t to — will indicate the rate of change of obliquity due to 
Clb 
the moon alone, to the sun alone, and to the sun and moon jointly. 
Then writing for P and Q their values, cos i and sin i, we have by (19) and (29), or 
by (30), 
