264 Mr. J. McCowan on the 



greater the greater the natural stream- velocity q : when there 

 is no stream-velocity q ebb will begin approximately at the 

 mean level h, but really somewhat below this, and more 

 notably when the tide is a high one. 



Again, by (34) and (42) , the duration of flood will be 



JT-- sin"' \ (43) 



where rj is given by (42) : therefore the duration of flood is 

 approximately 



±T _ T q-h/vo 



and that of ebb 



T--*^m (44) 



TqJ^o (45) 



Thus the durations of flood and ebb are very nearly equal 

 when there is no independent flow in the river, but the ebb 

 continues longer than the flow when there is an independent 

 flow q towards the sea, as was to be expected. 



All these results are in complete accordance with the 

 observed phenomena of the tides in rivers. Most of them, in 

 their approximate form, were given by Sir George B. Airy in 

 the article " On Tides and Waves," already referred to. His 

 approximation, however, led him to the further result that 

 the wave of the tide would subdivide as it advanced up the 

 river, so that at stations far enough up there would be two 

 times of high water in each period, or even three or more at 

 sufficiently distant stations. As a matter of fact there is such 

 a phenomenon of double tide in the upper parts of some 

 rivers, and this result was therefore taken as its explanation : 

 it is easy to see from (34), however, that there is really no 

 such subdivision, and that therefore the explanation of such a 

 peculiarity must be sought elsewhere. For (34) shows that 

 each point on the wave-surface may be regarded as advancing 

 up the river without change of elevation with a velocity 

 [3 \/ (<jz)— k~\, which depends only on its elevation, and 

 which is greater the greater the elevation. Hence the slope 

 at every point from the trough to the crest in front of the 

 wave, which near the mouth has the form of a curve of sines, 

 becomes greater, and at every point in the rear from crest to 

 trough it becomes less. Thus we can easily picture, or draw 

 if need be, the form of the advancing wave ; but without this, 

 however, the general considerations show that there can be no 

 subdivision, even though the wave were traced up to the point 



