Waves in Flowing Water. 57 



expression, remark that by (7) above and (34) Part III., we 

 have 



3Q _ A/J) (-ly+'cos^ _ A/D.N t - 

 l°g(l//.)"" ' Oi(l-b/D)sm*a t - 1-D/& • {W) ' 

 Now by putting <r=0 in Part III. (29) and (24), we find 



2N,=l-D/& (11). 



Hence, and by (10), (9) becomes 



:A/D (12). 



a Jo 



Denoting now, as before, by fy the height above mean level 

 from ridge to ridge, we find from (8), 



The comparison between this and (2) above, two different 

 expressions for the same quantity, (with, for simplicity, D = 1), 

 leads to the following remarkable theorem of pure analysis, 



. , . Iirx 



4 a . cos % 



*=" a 



i=1 ei+e-'-l.^-iei-e-i) 

 % 2irb v y 



~2 t J l-ftsin 2 *; ' 1-e"^ ' K ); 



where 



a denotes any real positive numeric; 



b denotes any numeric > 1 ; 



e denotes e 27r / a ; 



«,• denotes the numeric between zero and tt/2, 



which satisfies the equation \ (15). 



[(* — i) 73 "— a i] * an «,-— &=0; 

 f denotes (i— ^)7r— a,-; 

 # denotes any real positive numeric < a ; 



C a Ittx 

 The theorem (14) is easily verified by taking I dx . cos.; 



of both members. The first member of the result is obviously 

 2/|V +£"*•>— - • o~t (^ "^ «"/)]• ^ e se cond member, modified 



