AND ON THE REMOTE HISTORY OF THE EARTH. 
469 
n f C -f : - = }j sin i cos i( 1—f sin 2 i)E sin 2e+§ sin 3 i cos iE' sin e — f sin 3 iE" sin 2e 
dt 
sin i cos {(1 —§ sin 3 i)E sin 2e+f sin 3 i cos iE' sin e' —§ sin 3 iE'" sin 2e' 
U35) 
and by (33) and analogy with (19) and (29) 
ntt di tl , 
TTj dt 
— ^ sin 3 i cos iE sin 2e— sin i cos 3 iE’ sin e 
(36) 
The sum of these three values of — gives the total rate of change of obliquity due 
flu 
both to sun and moon, on the assumption that the three semi-diurnal terms may be 
grouped together, as also the three diurnal ones. 
It will be observed that the joint effect tends to counteract the separate effects ; this 
arises from the fact that, as far as regards the joint effect, the two disturbing bodies 
may be replaced by rings of matter concentric with the earth but oblique to the 
equator, and such a ring of matter would cause the obliquity to diminish, as was shown 
in the abstract of this paper (Proc. Roy. Soc., No. 191, 1878), by general considera¬ 
tions, must be the case. 
§ 10. The rate of tidal friction due to both sun and moon. 
The equation which gives the rate of retardation of the earth’s rotation is by (26) 
^- = —; it will however be more convenient henceforward to replace on by —n and 
to regard n as a variable, and to indicate by n 0 the value of n at the epoch from 
which the time is measured. 
Generally the suffix 0 to any symbol will indicate its value at the epoch. 
Then the equation of tidal friction may be written 
dfE\ — _i_ ■Ha! i 
dt\nj C n 0 C n 0 C n 0 
(37) 
Then by (22) and (34), in which the semi-diurnal and diurnal terms are grouped 
together, we have 
/ 9 • I 3 • 4, ’\ 771 * O I • 9, Vi ■> • 0 '\ 7,1/ / /0^'o\d®«r !1 ) 
-f )——= (cos^ if f snr i)E sm 2e+ sm J i(l — f sim i)E sm e = „ — 
0A) g - n l jjg s J n 2 e _|_ iS i n 3 i cos 2 gi n e ' 
tt, / Ln, 
y ■ ( 38 ) 
J 
