160 

 From equation (205): 



bX/^lK'-iaX/cy =djc (206) 



The integration of which gives: 



sin-i [iaX/c)/K]=ax/c^K', (207) 



in which K' is a second constant of integration. 

 From equation (207): 



X={cKfa) sin (ax/c+K') (208) 



This expression for A'' may be placed in the form: 



Z=Mcos (ax/c)-\-N sin (ax/c) (209) 



in which AI and N are constants. 



• Since the differential equation (202) for I^ is the same as that for JY", 

 the expression for Y is similarly: 



Y= P cos (ax/c) + Q sin (ax/c) , (210) 



in which P and Q are constants whose values are independent of those 

 of M and A^. 



The height of a component of the tide in the canal at a station dis- 

 tant x from the origin of distances, is then given by an equation in 

 the form: 



y = [M cos (ax/c) -\-N sin (ax/c)] cos at 



+ [P cos (ax/c)-\-Q sin (ax/c)] sin at (211) 



321. Expressions Jor components oj the current. — -The component of 

 the current due to the same component of the tide is, from equation 

 (193): 



v= V cos at-^Z sin at (212) 



From equations (200) and (210): 



V=(g/a)bY/dx=(g/a)[-(a/c)P sin (ax/c) + (a/c)Q cos (ax/c)] 



= (g/c) [Q cos (ax/c) — P sin (ax/c) ] (213) 



And from equations (200) and (211): 



Z^ — (g/a)bX/dx= — (g/a)[— (a/c)M sin (ax/c)-^(a/c) N cos (ax/c)] 

 = - (g/c)[N cos (ax/c) - M sin (ax/c) ] (214) 



