Waves in Flowing Water. 53 



where fy denotes height above mean level of the water 



at distance a? from the point over one of the 



ridges ; 

 A denotes profile-sectional area of one of the 



ridges ; 



yjr denotes 2irx/a ; f (3) 



e denotes £ 2,^ ^/ a • 

 M denotes the g/mXJ 2 of Part III. (6) to (18) 



or a/27rb ; 

 b denotes ~U 2 /g; 

 and D denotes the depth. 



Thus, in (2) we have an expression for the surface-effect of 

 an endless succession of equidistant ridges on the bottom. 

 We shall see presently that if the succession of ridges is finite, 

 the result expressed by (2) will not be approximated to by 

 increasing the number of ridges. The difference in the effect 

 of a million equidistant ridges from that of a million and one 

 equidistant ridges, in respect to the corrugations on the surface 

 of the fluid over any part of the series, may be as great as the 

 difference between the effects of a thousand and of a thousand 

 and one, or between the effects of ten and of eleven : and the 

 absolute effect of four, or six, or eight, may be sensibly the 

 same as, or may be greater than, or may be less than, the 

 effect of a million, in respect to the condition of the surface 

 over the space between the two middle ridges. The awkward- 

 ness of the consideration of infinity for our present case is 

 beautifully done away with, after the manner of Fourier, by 

 substituting for an "infinite canal" an "endless* canal," or 

 a canal forming a complete circuitf : a circular canal as we 

 may imagine it to be, although it might be curved, of any 

 form, provided only that, whether it be circular or not circular, 

 the radius of curvature at any point is very great compared 

 with the breadth of the canal. This condition is all that is 



* It is curious that the word " endless " should in common usage, and 

 especially in technology, have so different a meaning from "infinite." 

 Thus every one understands what is meant by an " endless cord." An 

 "infinite cord" means, in common language, an infinitely long cord — a 

 cord which has no limit to the greatness of its length. 



t A curious piece of illogical usage in mathematical language, according 

 to which an enclosing curve is called a " closed curve," must henceforth 

 be absolutely avoided. It has already led to endless trouble in electrical 

 nomenclature, according to which, in common language, an electric cir- 

 cuit is said to be closed when a current can pass through it, and to be open 

 when a current cannot pass through it. I believe all, or almost all, English 

 writers on electrical subjects have been guilty of this absurdity. I doubt 

 whether any one of them would say a road round a park is open when a 

 gate on it is closed, and is closed when every gate on it is open. 



