AND ON THE REMOTE HISTORY OP THE EARTH. 481 
Then it for the moment we call the right-hand of this equation k, we have 
fl —Integrating a second time, we find that the moon has fallen behind her 
proper place m her orbit —--— seconds of arc m the time t. Put t equal a 
century, and substitute for k, and it will then be found that the moon lags in a century 
630‘7i? sin 2e+108-6-E''sin e — 7"042i?" sin 2e" seconds of arc . . (60) 
But it was shown in Section 13 (48) that the moon, if unaffected by tidal reaction, 
would have been apparently accelerated 1043 - 3i?sin 2e+232 , 5jE" sin e seconds of arc 
in a century. 
Hence taking the difference of these two, we find that there is an apparent 
acceleration of the moon’s motion of 
412 , 6i? sin 2e+123 , 9£’ / sin e / + 7 , 042i?' / sin 2e".(61) 
seconds of arc in a century. 
Now according to Adams and Delaunay, there is at the present time an unex¬ 
plained acceleration of the moon’s motion of about 4" in a century. For the present 
I will assume that the whole of this 4" is due to the bodily tidal friction and reaction, 
leaving nothing to be accounted for by ocean tidal friction and reaction, to which the 
whole has hitherto been attributed. Then we must have 
412-6£sin 2e+123-9A'sin F+7-042A" sin 2e" = 4 .... (62) 
This equation gives a relation which must subsist between the heights E, E ', E", of 
the semi-diurnal, diurnal, and fortnightly bodily tides, and their retardations e, e', e", 
in order that the observed amount of tidal friction may not be exceeded. But no 
further deduction can be made, without some assumption as to the nature of the 
matter constituting the earth. 
I shall first assume then that the matter is purely viscous, so that E— cos 2e, 
E'= cos e, E" — cos 2e", and tan 2e=~- 1 , tan e'=-, tan The equation then 
P P P 
becomes 
412-6 sin 4e+123’9 sin 2e' + 7-Q42 sin 4e"=8.(63) 
If the values of e, e, e" be substituted, we get an equation of the sixth degree for p, 
but it will not be necessary to form this equation, because the question may be more 
simply treated by the following approximation. 
