Waves in Flowing Water* 55 



But now let us consider the case of M(e* — e"')/(e' + e~ { ) as 

 far as possible from being an integer ; that is to say, 



M(^--e- i )/(e i +e' i )=j + i .... (5), 



where j is an integer. For all values of i less than j + 1 the 

 denominator of (2) is clearly negative, with increasing abso- 

 lute values up to i—y, and for all values of i greater than^ it 

 is positive, with decreasing values from i=j+l to i = oo. 

 Thus the absolute magnitudes of the coefficients of cos iyfr in 

 the successive terms of the series from the beginning are nega- 

 tive, with increasing absolute values up to i=j ; and after that 

 positive,with decreasing values converging ultimately according 

 to the ratio e~K Remembering that e = e 47rD/a , we see that the 

 convergence is sluggish when a, the distance from ridge to 

 ridge (or the length of the circuit in the case of an endless 

 canal with one ridge only,) is very large in comparison with 

 the depth ; but that when a is less than the depth, or not 

 more than five or ten times the depth (an exceedingly inte- 

 resting class of cases), the convergence is very rapid. 



We shall find presently, however, another solution still 

 more convergent, much more convergent indeed for the 

 greater part of the configuration, whatever be the ratio of D 

 to a ; a solution which is highly convergent in every case 

 except for values of x considerably smaller than the depth. 

 The calculation for these small values of x is necessary to 

 give the shape of the water-surface at distances on each side 

 of the vertical through the ridge small in comparison with 

 the depth : for this purpose, and for this purpose only, is the 

 solution (2) indispensable. For investigating all other parts 

 of the configuration the new solution is much more convenient, 

 and involves, on the whole, very much less of arithmetical 

 labour. It is found by summation from the solution of the 

 single-ridge problem given in Part III. (40), (41), as follows. 



Let the whole number of ridges be^+/ + l, and let it be 

 required to find the shape of the surface between the verticals 

 through ridges numbers j and j -f 1 . Take the origin of the 

 coordinate x in the vertical through number^' ridge, and let 

 number ^' + 1 be on the positive side of it. The solution will 

 be found by adding to the solution (40) Part III., j solutions 

 differing from (40) only in having respectively x -f a, x + 2a, 

 . . . , x +ja substituted for x ; and f solutions each the same 

 as (40) Part III., but having — x + a, —x + 2a,.., —x+fa, 

 substituted for x. Thus, denoting by S the sum of the effects 

 of the j +f + 1 single ridges, we find 



«J-n W-f?X la ^-W*° }. . . (6); 



