346 TIDAL WAVES. [CHAP. VIII 



As in Art. 190, this last result greatly simplifies the equation 

 of continuity. In the present case we find without difficulty 



d(hv)} 

 dt v\ RdO day J" 



207. It is important to notice that these equations involve no 

 assumptions beyond those expressly laid down ; in particular, 

 there is no restriction as to the ellipticity of the meridian, which 

 may be of any degree of oblateness. 



In order, however, to simplify the question as far as possible, 

 without sacrificing any of its essential features, we will now take 

 advantage of the circumstance that in the actual case of the earth 

 the ellipticity is a small quantity, being in fact comparable with 

 the ratio (n*a/g) of centrifugal force to gravity at the equator, which 

 is known to be about ^g. Subject to an error of this order of 

 magnitude, we may put R = a, TV = a sin 0, g = const., where a is 

 the earth's mean radius. We thus obtain* 



du n a d /v , -r,. 



-J- - 2nv cos = --=-,(- f), 

 at a au 



_ ) d(hv) 



~ ~~ ~ 



dt a sin 



){ 

 dt~ ~^in~0 dd~ da>~) ' 



this last equation being identical with Art. 190 (1). 



Two conclusions of some interest in connection with our previous work 

 follow at once from the form of the equations (1). In the first place, if u, V 

 denote the velocities along and perpendicular to any horizontal direction s, we 

 easily find, by transformation of coordinates 



-v cos = - 



In the case of a narrow canal, the transverse velocity v is zero, and the 

 equation (i) takes the same form as in the case of no rotation ; this has 

 been assumed by anticipation in Art. 180. The only effect of the rotation 

 in such cases is to produce a slight slope of the wave-crests and furrows 

 in the direction across the canal, as investigated in Art. 201. 



Again, by comparison of (1) with Art. 200 (7), we see that the oscillations 

 of a sheet of water of relatively small dimensions, in colatitude 6, will take 

 place according to the same laws as those of a plane sheet rotating about 

 a normal to its plane with angular velocity n cos 6. 

 * Laplace, 1. c. ante p. 343. 



