4 
THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 
distances of those luminaries at a certain time, anterior to the time of the tide by the 
interval occupied in the transmission of the tide along the channel. Let this interval 
of time be called the Retroposition of the theoretical tide in time. It is what, on former 
occasions, I have called the “ age of the tide." 
6. This phraseology being adopted, the phenomena of the Liverpool tides may be 
expressed as follows. 
(1.) The effects which the changes of the Moons force produce upon the Tides, are 
the same as the effects which those changes would produce upon a Retroposited Equili- 
brium-tide. 
(2.) The Retroposition of the Tide in longitude is affected by small changes , ivhich 
changes are proportional to the variations in the moons tidal force. 
(3.) The Retroposition of the Tide in time is also affected by small changes, which 
changes depend on the variations of the moons force. 
7 . The former of these propositions is proved by the verification of the formulae 
already mentioned, since these agree with the formulae for equilibrium-tides, except 
in the circumstance of having <p — a for <p. Now this difference is equivalent to a re- 
troposition of the tide in time, of such magnitude that, during this time, the distance 
of the sun and moon is changed from <p — a to <p. If a be l h 15 ra , as collected from 
the law of the times, the retroposition in time is the time requisite for the moon to 
75 
increase its right ascension from the sun by l' 1 15 m ; that is, it is ^ days nearly, or I 
day 13^ hours. The tide at Liverpool agrees nearly with an equilibrium-tide pro- 
duced in the southern ocean, 3 7\ hours previously to the moon’s transit at that port, 
and transmitted thither unchanged. 
8. The second of the above propositions is proved by tracing the effect of changes 
of lunar parallax and declination upon the results compared with the above formula 
for the times and heights, namely, 
tan 2 (O' — X 1 ) 
h sin 2 (<p — a) 
h 1 + h cos 2 (<p — a) 
(a.) 
y = {h' 2 + h 2 + 2 h h' cos 2 (<p — a)} 
(b.) 
By the equilibrium-theory, the change which would be produced by any alteration of 
the moon’s force would correspond to the effect of an alteration in the value of h!, the 
amount of the lunar tide. It appears from the examination of the observations, that 
this change takes place in fact, but that we must also suppose a change in X 1 in order 
that the formula (a.) may represent the observed intervals of time. This change in 
X', the retroposition of the tide in longitude, is 
2 m "5 ( p — 37) for parallax p minutes. (See Art. 15.) 
84 m sin 2 S for declination l. (Art. 21 .) 
Now, by the theory, the effect of a change in the moon’s parallax on the equilibrium- 
tide is as the change of parallax; and the effect of the moon’s declination is a change 
