576 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
effects of inertia are supposed to be small. Hence y must be a small angle, and there 
will not be much error in putting 
X=t Z 5 w“ s i n e cos e, and sec y= 1- 
Then we have for the lag of the tide ( e+sin e cos e), and for its height 
\ 9 
cos + cos 3 ^. 
Let 7] be the lag, then 
o 
_„ L 79 
whence 
Also 
and 
Hence (61) becomes 
where 
>/ = e+ i -50 - sm e cos e, 
e=y — — sin rj cos rj very nearly. 
cos e= cos 7]( sin' rj ), 
cos e l+^- cos- e = cos A l+iVt) 7 
a S 
a q 
n o T 
- = Z cos m 1 +T 51 J 7 , COS (vt — 7)). 
‘1 
r 
(62) 
7] — T 5 o“ sm 7] cos 7] — arc tan - 
/ 19vv\^ 
\2gawJ J 
This is probably the simplest form in which the result of the second approximation 
may be stated. 
From it we see that with a given lag, the height of tide is a little greater than in 
the theories used in the two previous papers; and that for a given frequency of tide 
the lag is a little greater than was supposed. 
The whole investigation of the precession of the viscous spheroid was based on the 
approximate theory of tides, when inertia is neglected. It will be well, therefore, to 
examine how far the present results will modify the conclusions there arrived at. It 
would, however, occupy too much space to recapitulate the methods employed, and 
therefore the following discussion will only be intelligible, when read in conjunction 
with that paper. 
The couples on the earth, caused by the attraction of the disturbing bodies on the 
tidal protuberance, were found to be expressible by the sum of a number of terms, 
