﻿740 ProE. H. Lamb and Miss L. Swain on 



The calculations which follow will serve to illustrate the 

 foregoing remarks. The formula were worked out originally 

 in response to an inquiry addressed to one of the writers from 

 abroad, and in ignorance, or more probably in forgetfulness 

 of the fact that the matter had already been treated to some 

 extent by Airy, and referred to by Hough. 



We consider the case of an equatorial canal of uniform 

 depth //, the moon being supposed to revolve in the plane of 

 the equator. If 6 denote longitude measured eastwards 

 from a fixed meridian, and nt the hour-angle of the moon 

 west of this meridian, the dynamical equation is of the form 



P = JP-/sin2(n*+*), ... (8) 



where a is the earth's radius, c 2 =gh, and f denotes hori- 

 zontal displacement eastwards*. 



For an equilibrium theory we neglect the term ^^fat 2 . 

 If the origin of 6 be taken at the centre of the canal, we find 



f=-«^-< sin 2wi cos 2*4— cos 2nt sin 2ot— sin 2(nt+0) > ; 



... (9) 



for this expression satisfies the differential equation, and 

 makes f = at the ends {6= + a). For the surface-elevation 

 we have 



7idf Itt r w M , m sin 2^ ") 



. . . no) 



where H—fag. This quantity H measures, on the equili- 

 brium theory, the maximum range of the tide in the case of 

 an ocean covering the whole earth |. 

 At the centre (0 = 0) we have 



1 / sin 2 a \ 



7 ? =iHcos2^M-^^Y . . . (11) 



If a. be small, the range here is very small, but there is not 

 a node in the strict sense of the term. The times of high 

 water coincide with the transits of the moon and antimoon. 

 At the ends (0= ±a), we find 



1 TT f/ -. sin4a\ n/ , N 1— cos 4a . ... ."] 



77= 2 H \( 1 "~4a _ ) COs2(nf±a)=f: 4a Sin 2(^±a)j 



= |H.BoCos2(wt±a+^ ) 5 (12) 



if -d a i i sin 4a .p . 1— cos 4a 



R cos2</> =1 j—, Ro sm2 r>o= Ya. ' ' ( 13 ) 



* 'Hydrodynamics," Art. ISO. 

 t 'Hydrodynamics,' Art. 179. 



