256 THE REV. W. WHEWELL’S RESEARCHES ON THE TIDES. (SERIES XII.) 
half-day, the height of this point above a horizontal line will represent the height of 
the surface of the water, supposing the velocity and the radius to be duly adjusted. 
But the radius depends upon the total rise of the surface, and the velocity depends 
on the length of a tidal day, and both these quantities are different at different 
periods of a lunation. Hence our line of sines will, even theoretically, require some 
modification. 
Let the height of the surface above a given zero line be 
7 I 7 <i ' nt 
y = k h cos 
where r is the length of the tidal half-day, and t the time from high water ; 2 h the 
rise from low water to high water. 
The quantity r will (according to theory) differ at different periods of the semilu- 
nation. The law of its change is determined as follows : 
Let p be the R.A. of the moon minus a certain constant, and 0 the time at which the 
high water (or low water) follows the moon’s transit, minus a certain other constant ; 
c the ratio of the solar to the lunar tide. Then by the equilibrium theory we have 
tan 2 0 = 
c sin 2 <p 
1 c cos 2 <£> 
Also if <p’, O’, be the values of <p, 0, at the next tide, 
, c sin 2 <$’ 
tan 2 0 = — 
1 + c cos 2 
Hence tan (2 O' - 2 0) - - C (sin 2 ^ ~ sin 2 ^ ± sin 2 ~ ^ 
Hence tan {20-20) — - 1 + C ( C0S 2^ + cos2<p) + c 2 
— _ e cos (<?' + <p) + c 2 cos (?'-?) . . , _ , 
1+2 c cos (<p r -j- $) cos (<p' — <p) + c 3 
But the arcs 2 0' — 2 0 and q>' — <p are small, and very nearly as their tangents and 
sines. Also cos (<p' — ■ <p) = 1, nearly, and cos ( <p ' -f <p) = cos 2 <p, nearly. 
c cos 2 <p + c 2 
Hence O' — 0 — — (p' — <p) 
1+2 c cos 2 <p + c 
. 2 ’ 
i , 1 + c cos 2 <p , , . 
whence O’ — 0 + d — <p = | , 0 - Q . , a (<P — ?>)• 
Now the former tide happens <p + 0 after the sun’s transit, and the latter tide <p' 4* O' 
after the sun’s next transit ; therefore the length of the tidal half-day is 12 h + ^ + 
— 0 — <p. Also <p’ — <p is constant, being always very nearly 24 minutes. Hence 
the tidal half-day will be least and greatest when the above fraction is so ; that is, 
when 2 <p is 0 and ir. The tidal half-day is therefore least at spring tides, when it is 
4 >. 
12 h + 
1 + c 
and greatest at neap tides, when it is 
12 h 
<P' -<P 
1 — c 
