179-181] CANAL COINCIDENT WITH MERIDIAN. 291 



Substituting in the equation (5) of Art. 174, and solving, we find 



?; = jffcos2- + j f ,, cos2-.cos2(X + 6) .. .. (3). 

 4 a 4 c 2 -n 2 a 2 a 



The first term represents a change of mean level to the extent 



(4). 



The fluctuations above and below the disturbed mean level 

 are given by the second term in (3). This represents a semi 

 diurnal tide ; and we notice that if, as in the actual case of the 

 earth, c be less than n a, there will be high water in latitudes 

 above 45, and low water in latitudes below 45, when the moon 

 is in the meridian of the canal, and vice versa when the moon 

 is 90 from that meridian. These circumstances would be all 

 reversed if c were greater than na. 



When the moon is not on the equator, but has a given declination, 

 the mean level, as indicated by the term corresponding to (4), has a coefficient 

 depending on the declination, and the consequent variations in it indicate a 

 fortnightly (or, in the case of the sun, a semi-annual) tide. There is also 

 introduced a diurnal tide whose sign depends on the declination. The reader 

 will have no difficulty in examining these points, by means of the general 

 value of ii given in the Appendix. 



Wave-Motion in a Canal of Variable Section. 



181. When the section (S, say) of the canal is not uniform, 

 but varies gradually from point to point, the equation of con 

 tinuity is, as in Art. 166 (iv), 



where b denotes the breadth at the surface. If h denote the 

 mean depth over the width b, we have S = bh, and therefore 



where h, b are now functions of x. 



The dynamical equation has the same form as before, viz. 



#_ a ^i 



dP~ ~ g dx ........................... ( 



192 



