AND ON THE REMOTE HISTORY OF THE EARTH. 
497 
2 7Z" 
tan 2e= —, tan e'=- for tlie semi-diurnal and diurnal tides respectively, 
annual tide will be neglected. 
The semi- 
Then if the viscosity so varies that all the e’s are always small, and if we put —=\, 
we have 
sin 4ej sin 4e 3 
• A 1 ””” A.J 
sm 4e 
sin 2e\ 
sin 4e 
: 1 +X) AA =X 
sin 4e 
sin 2e' 
— l _ \ — l 
^ sin 4e ^ 
sin 4e 
sin 2 eh 
sin 4e 
=l+x 
(76) 
By means of these equations we may expresss all the sines of the e’s in terms of 
sin 4e. 
Then, remembering that the spheroid is viscous, and that therefore Ep= cos 2e 1} 
E\ = cos e\, &c., we have by Sections 4 and 7, equations (16) and (29), 
s ^- n <f) sin 4e —\pcf sin 4e 3 —-fp 3 ^ 3 sin 4e" 
+iP 5( Z(p 2 + 3g f3 ) sin 2 e' 1 —^pq(p 2 —q 2 ) 3 sin 2e'—-Ipp^Spd-l-^ 2 ) sin 2eh] . (77) 
- jr — —[ijp 8 sin 4e 1 + 2ph/ 4 ' sin 4e+^ 8 sin 4e 3 
dt ga 0 ' 
+p 6 # 3 sin 2e'j -j -p 2 q 2 (p 2 —q 2 ) 2 sin 2e' -\-p~q° sin 2e' 3 ] . (78) 
And by (57), Section 14, 
o 
— [-|p 8 sin 4e x —sin 4e 3 — Sp^q* sin 4e ,/ + 2p 6 (j' 3 sin 2e] —2 p 2 q 6 sin 2e / 3 ] 
(79) 
The first two of these equations only refer to the action of the moon on the lunar 
tides, but the last is the same whether there be solar tides or not. 
Then if we substitute from (76) for all the e’s in terms of sin 4e, and introduce 
cos i=P=p 2 — q 2 , sin i—Q—2pq, we find on reduction 
d'ljii 
1 
dt N g % 0 
sin4e[^PQ— 
dN m 
dt 
^£ _sin4€ [ 1_ ^ 3 ” xp ] > 
-(M t 
(80) 
