OF THE TIDES IN THE PORT OF LONDON. 35 



In this expression, the numbers being- feet, / at the London Docks is 21-33 above 

 the origin of the measures used in the Tables. 



*' = 17 cos (2 ?) - 5 1°) — -23 sin (4 (p - 30°) 



/== (p-V) {'\7 + bcos2<p] 



q' = (sin2 A — sin2 h) {73 — c? cos 2 (?) — 45)} 



q = m sin 2 {<p — {jb) -\- n sin {(p — v), 

 b, d being not clearly shown by the London observations to be constant terms ; and 

 m, n, as before, being small, and their determination for the present omitted. 



Chap. III. Comparison of the preceding Results with the Theory. 



I shall now compare the preceding results with the theory of Daniel Bernoulli, 

 according to which the waters of the ocean assume nearly the form in which they 

 would be in equilibrium under the action of the sun and moon ; and a supposition 

 being made that the pole of the fluid spheroid follows the pole of the spheroid of 

 equilibrium at a certain distance (namely, at an hour-angle V), and that the equili- 

 brium corresponds to the configuration of the sun and moon, not at the moment of 

 the tide, but at a previous moment, at which the right ascension of the moon was 

 less by a quantity a. 



I take this theory rather than that of Laplace, not only because of the difficulty 

 and labour of the comparison in the latter case, but also because the hypothesis on 

 which Laplace's solution proceeds appears to me likely to affect the results, so as to 

 make them differ altogethei* from those of the real case ; and because the assumption, 

 by means of which his solution is obtained, appears to me to be very insecure. 



According to the theory of Bernoulli, we have 



^/A i\ /isin2(d> — a) ,, . 



tan2(^-K')= ^- + ^eosa(^-») > (!•) 



where & is the hour-angle corresponding to the place of high water measured from the 

 moon, (p the hour-angle of the moon from the sun, h, H the heights of the solar and 

 lunar tides, X' the hour-angle by which the tide follows the pole of equilibrium, a the 

 retardation, or difference of right ascension of the moon due to the age of the tide. 



Neglecting the effects of parallax and declination, this expression gives the law of 

 the semimenstrual inequality ; and this, as we have already said, agrees very clearly 



with observation, assuming proper values of ^5 and of a. 



But we find here some circumstances in which the theory and observation 



are difficult to reconcile. The value of -r-, the ratio of the lunar and solar tide, ought 



to be the same at all places. We find, however, that the Brest observations give 2-6167, 

 while the London observations require it to be 2*9887 ; and other places give values 

 still more different. 



Also the diffierences of the value of a for different places might be supposed to 



f2 



