Waves in Flowing Water. 529 



solution promised in the last sentence of 

 Part II. The demonstration that this is the 

 practically unique solution for inviscid 

 water flowing in a canal with a single 

 ridge, and the explanation of how any other 

 state of motion, such, for example, as that 

 represented by (46) with any value of C and 

 C, but given to the water throughout only 

 a finite distance on each side of the ridge, 

 settles into the permanent steady motion 

 represented by (48), must be reserved for 

 Part IV., which I hope will appear in the 

 January number. 



Meantime the accompanying diagram 

 represents by two curves two cases of the 

 solution (46) for the particular value 2*456 

 of D/b ; that is to say, for velocity = '6381 

 of the critical velocity v'y.D. The faint 

 curve represents the solution (39) (41), or, 

 which is the same, (46) with C = and 

 C / =0. The heavy curve represents the 

 practical solution (48). These curves were 

 drawn from calculations of a periodic 

 solution, according to the first of the two 

 methods indicated above, before I had 

 found the analytical solution (39) by which 

 the desired result could have been arrived 

 at with much less labour. The faint curve 

 was drawn first by direct calculation from 

 the periodic solution : the letters \ I, J /, 

 ~~ih ~ih & c -> show on the two sides of 

 one ridge quarters of the distance from 

 ridge to ridge in the periodic solution, one 

 of the ridges being in the middle of the 

 diagram. The heavy curve is found by 

 adding to the ordinates of the faint curve 

 the ordinates of a curve of sines, found by 

 trial to as nearly as possible annul on the 

 one side, and to double on the other side, 

 the ordinates of the original curve. How 

 nearly perfect was the annulment on the 

 one side and the doubling on the other is 

 illustrated by the small-scale diagram an- 

 nexed (fig. 3) , which has been drawn by 

 the engraver from a six times larger copy. 

 How nearly perfect the annulment and the 



