﻿a Tidal Problem. 741 



Hero R is the ratio of the range of the tide to the quantity 

 H, and </> denotes the hour-angle of the moon W. or! the 

 meridian when there is high water at the eastern end of the 

 canal ; it is also the hour-angle E. of the meridian when 

 there is high water at the western end *. When a is small 

 we have 



H = 2k, ^=-1^+3*, . . . (14) 



approximately. 



The values of R and <f> for a series of values of a. ranging 



from to - 7r are given in the table at the end of this 



paper, on the assumption that h= 10820 feet. 



When the inertia of the water is taken into account, 

 we have 



f = a 7 / Txo sin 2(nt+6) - -r— A \ sin 2(nt + a) sin 2m{6 f a) 



4z{m' — l)c z L sin 4raa [_ v v ' 



— sin 2(nt~x) sin 2m(0 — a)\ , . (15) 



where m=na/c. For this satisfies (8), and vanishes for 

 0=+«t' Hence 



v =- h M 



V a~dd 



1 H r . . m f 



= ~~2^ — r cos2(w* + 0)- . \ sin2(n* + a)cos2»*r0 v «) 



Lm — ±L sm -kino. [_ ; v ' 



— sin 2{nt-u) cos 2m(<9-a)"|l. . (16) 

 An equivalent form is 



1 TT r r 



v = " 9 7^~T cos 2 ('^ + °) ~ ^-r— 1 cos 2 (^ + »'*) sIn -< "■ + 1)* 

 ^ m — 1 L sin4ma L 



— cos2(W — m0) sin 2(m— Tall. (17) 



j -- 

 If we imagine m to tend to the limit we obtain the 

 formula (12) of the equilibrium theory. It may be noticed 

 that the expressions do not become infinite for m— >1, as in 

 the case of a canal encircling the globe. In all cases, bow- 

 ever, which are at all comparable with oceanic conditions, m 

 is considerably greater than unity. 



* The phase-difference is 2$ . This angle, reckoned in degrees from 

 to 3()0°, is called by Darwin and Baird the " lag " of the tide, Proc 

 Koy. Soc. vol. xxxix. p. 18o (1885). 



f Cf. Airy, « Tides and Waves.' Art. 301. 



