OF THE TIDES IN THE PORT OF LONDON. 



37 



The expression which we obtained from observation was 

 P' = (P-/')(B + Bsin2(9-/3)), 

 of which the second term agrees in form with the one given by the theory, except 

 that the angle ^ is different from a ; but the first term should not occur according to 

 the theory as hitherto stated. 



2. Again, for the effects of lunar declination on the time of high water. 



If A:' be the value of h' when the place of observation and the pole of the lunar 

 tide spheroid are both in the equator, and if g be the difference of declination of the 

 place and the pole in any other situation, we shall have A' = A:' cos' £ nearly. 



In the course of a tide-day there are two tides, corresponding to two positions of 

 the tidal spheroid ; and if / be the latitude of the place, I the declination of the moon, 

 the two corresponding differences of declination will be / — ^ and l-\-l, the pole of 

 the spheroid being supposed to have the same declination as the moon has at the 

 moment of the origin of the tide (that is, when the moon's right ascension was less 

 by a than it is at the moment of the tide). 



Then, in the first case, 



H = k^ cos* g = J<f cos* (/ — ^) = k* (cos / cos 5 — sin / sin ly 

 = k' {cos* / — 2 sin / cos / sin ^ cos 5 — (cos* / — sin* /) sin* h} 

 = A;' cos* / — 1 A;' sin 2 /sin 2 ^ — A;' cos 2 / sin*$. 



In the second case, similarly, 



h' = k' cos* / 4- i A;' sin 2 / sin 2 $ - A:' cos 2 / sin* i. 



In order to find the effect of the declination upon each tide, we should put for 

 ^ h' the quantities — J A;' sin 2 / sin 2 S — A;' cos 2 / sin* I, and + J A;' sin 2 / sin 2 J 

 — A:' cos 2 / sin* § respectively. 



Thus, according to the theory, the effect of declination on the two tides of the same 

 day should be different. This difference is very much modified by the circumstances 

 in which the actual state of the ocean differs from the theoretical state : the differ- 

 ence of the diurnal tides may, however, be detected in the observations at most places 

 of the earth's surface, perhaps at almost all. But there are peculiar circumstances 

 in the port of London which affect this difference, and obliterate it : the tide at 

 London is composed of two tides, which differ by half a day from each other, and 

 hence the difference of the two semidiurnal tides disappears altogether. Therefore, 

 instead of the effects of declination on the two semidiurnal tides, we must take the 

 mean of these effects, which is — Jc cos 2 / sin* ^. 



Hence if Q' represent the effect of lunar declination on the time of high water, we 

 have by equation (2.) (substituting — A;' cos 2 ^ sin* ^ for I h, and putting the arc for 

 tan 2 (^' — X') ), 



2Q' = 



.~-z-, -T -7- C03 2 / sm 6. 



sin 2{f — a) h 



In this expression we have A;' cos' / for h' in the value of tan S' ; but it is clear that 



