ON v;aVES,- 343 



V the volume of fluid constituting the wave ; 



ff the measure of gravity ; 



a the horizontal range of translation ; 



X the wave length or amplitude ; 



an arc — , m being an arbitrary number ; 

 m 



^ the arc whose sine = ^ versed sine of 6 ; 



TT the number 3- 141 6 ; 



*' the circumference of an ellipse whose axes are given ; 



X and y horizontal and vertical ordinates of wave-curve ; 



x' and y' horizontal and vertical ordinates of translation-path ; 



h' the height of a particle in repose, above the bottom of the channel. 

 Then we have 

 (1.) For velocity of wave transmission, 



c= \^g(h+k) B. 



= x^ffh nearly, when h is small A. 



(2.) For the wave length, 



X='2Trh—a F. 



^=21:11 nearly, when k is small E. 



(3.) For the range of translation, 



a= ^- always, 

 bh ■' 



= «k when k is small, = 2 A nearly when k is large L. 

 (4.) For the wave form, 



x=zhd—x':=hd, when k is small 



t/=^k. versin d G' 



(5.) For the path of translation, 



x'=a versin ^ \ ^, 



y'=^k versin 9 J 



and below the surface at A', w'=-—. versin P. 



(6.) The limits of the value of k are as follows : — 



Inferior limit ^=0, and k=h superior limit K. 



(7.) The range of vertical motion of a particle during translation being 

 y=zk at the surface ; the range of vertical motion of any other particle at the 

 height h above the bottom is 



y'=^~k N'. 



Geometrical Representation of the Wave of the First Order. — These data 

 enable to approximate to the exact conception of the motions of the wave 

 particles, and the relations which the wave form and the particle path bear to 

 each other. We may thus construct a geometrical representation of the wave 

 motion, which, however, is to be carefully distinguished from a physical de- 

 termination of its phaenomena. 



Let us then endeavour to follow the motion of a given particle on the sur- 

 face of the fluid during the wave form transmission. 



Let us talie D E for the depth of the fluid. (Plate LIL fig. 3.) 



Let us take C D for the height of the wave. 



Let us mark off rf D rf' = the circumference of the circle of which D E is 

 the radius = 6'2S32 X D E. Let also semicircles be described on c d and on 

 c'd' each equal to C D. Let the semicircles cd and c' d' and the distances 



