290 TIDAL WAVES. [CHAP. VIII 



where a is the hour-angle of the moon from the meridian of P. For 

 simplicity, we will neglect the moon s motion in her orbit in comparison with 

 the earth s angular velocity of rotation (n) ; thus we put 



, 



asm 6 



and treat A as constant. The resulting expression for the component X of 

 the disturbing force is found to be 



x - -si- 



We thence obtain 



. .sin 20 sin 2 A cos 



sin 2 ^ sin 2 A cos 2 (nt+ 



5-9 - 9 9 o ,, . .. 



z c 2 - n 2 a? sin 2 6 \ a sin 6 



The first term gives a diurnal tide of period ^irjn ; this vanishes and 

 changes sign when the moon crosses the equator, i.e. twice a month. The 

 second term represents a semidiurnal tide of period TT/U, whose amplitude is 

 now less than before in the ratio of sin 2 A to 1. 



180. In the case of a canal coincident with a meridian we 

 should have to take account of the fact that the undisturbed 

 figure of the free surface is one of relative equilibrium under 

 gravity and centrifugal force, and is therefore not exactly circular. 

 We shall have occasion later on to treat the question of displace 

 ments relative to a rotating globe somewhat carefully ; for the 

 present we will assume by anticipation that in a narrow canal the 

 disturbances are sensibly the same as if the earth were at rest, 

 and the disturbing body were to revolve round it with the proper 

 relative motion. 



If the moon be supposed to move in the plane of the equator, 

 the hour-angle from the meridian of the canal may be denoted by 

 n t + e, and if as be the distance of any point P on the canal from 

 the equator, we find 



CT 



cos ^ = cos - . cos (n t -f e) .................. (1). 



Hence 



X = j- = - /sin 2 - . cos 2 (n t -f e) 

 dx a 



= --J/sm2-.{l+cos2(X-f e)} ............ (2). 



a 



