CONSTRUCTION FOR VELOCITY AND HEIGHT. 13 



which is parallel and equal to it. Therefore the whole 

 retardation since leaving B is proportional to the sum of 



all the abscissae pp' that is, to Bp'. It is H - . 



T ZTT 



This represents the defect from the greatest eastward velo- 

 city ; and after passing its mean value at the middle point 

 s it represents a velocity which, relatively to the earth, is 

 westerly. The velocity of the current relatively to the 

 earth is represented hy ps. 



We shall now show that the height of the tide at ' 

 ahove its lowest point is also proportional to Bp'. 



If at any point in the supposed canal a thin section be 

 taken, the quantity of water entering this section in a given 

 time is proportional to the product of the depth and the 

 velocity. If the water flows in a little more rapidly than 

 it flows out, it is clear that the increase in the quantity 

 contained in the section, and therefore the increase in depth, 

 will be proportional to the difference between these two 



i i j xu /diff. of vel. x depth \ 



velocities and to the whole depth ~ - }. 



\ length of section / 



This holds as long as the change is small compared with 

 the whole depth. If this be supposed uniform throughout 

 the canal, the increase in it (that is, in the height of the 

 tide) at a' is therefore proportional to the retardation ; 

 and since the tide began to rise at B, where the velocity 

 began to diminish, it follows that Bp' is also proportional 

 to the height of the tide at a above its lowest point. 



It is easy to deduce from this construction the cor- 

 responding formulae. For, if OB = r, we have 



And since sB is proportional to the mean height, the defect 



