AND ON THE REMOTE HISTORY OF THE EARTH. 
Then by the theory of bodily tides as developed in my last paper 
471 
E— cos 2 e, E'— cos e, E"— cos 2 e", E'" — cos 2 e" 
tan 2e=—, tan 2 e "= + tan 2 e '" = 2 ^ 
P P P P 
(40) 
Rigorously, we should add to these 
Ei= cos 2 e 1} -Eo— cos 2 e 2 , cos e\, E ' 2 = cos eh 
, „ 2 (to-/ 2 ) , „ 2 (to + / 2 ) , , »- 2/2 , , n + 2/2 
tan 2e,= -, tan 2 e 0 =-, tan e , =-. tan e 2 = 
P P P P 
(40') 
But for the present we classify the three semi-diurnal tides together, as also the 
three diurnal ones. 
Then we have 
■- = [£ sin 7 cos 7(1 —f sin 3 1 ) sin 4e+f sin 3 7 cos 7 sin 2 e'](n 3 +?q 2 ) — +6 sin 3 7 sin 4e"ir 
— sin 3 7 sin 4e"u~ — ( 4 sin 3 7 cos 7 sin 4e+^ sin 7 cos 3 7 sin 2e)uu / . 
Now 
\ sin 7 cos 7(1 —f sin 3 7) = -+- sin 27(5 + 3 cos 27) = ^(5 sin 27+f sin 47) 
f sin 3 7 cos 7= 3 + sin 27(1 — cos 27) = - 6 3 4 -(2 sin 27— sin 47) 
sin 3 7= q\( 3 sin 7— sin 37), \ sin 3 7 cos 7=- 6 - 4 -(2 sin 27— sin 47) 
| sin 7 cos 3 i= l sin 27(1 + cos 27)=- 6 - 4 (2 sin 27+ sin 47). 
If these transformations be introduced, the equation for — may be written 
G 4~ — — 9(u z sin 4e" + M 3 sin 4e"') sin 7+3 (ift sin 4e" + « 3 sin 4e") sin 37 j 
at 
+ [ (5 sin 4e+6 sin 2e')(ir+w / 3 ) — (4 sin 4e + 8 sin 2e')im / ]sin 27 r* • (^1) 
+ [(f sin 4e —3 sin 2 e , )(w 3 +w / 3 ) + (2 sin 4e—4 sin 2 e')mq] sin 47 J 
Then substituting for u and u t their numerical values (39), and omitting the term 
depending on the semi-annual tide as unimportant, I find 
64—= —5‘9378 sin 4e" sin 7+1’9793 sin 4e' sin 37" 
dt 
+ {2’7846 sin 4e+2'3611 sin 2e'} sin 27 
+ {1*8159 sin 4e —3'G317 sin 2e'j sin 47 
3 P 
(42) 
MDCCCLXXIX, 
