1874.] Uniform Wave of Oscillation. 353 



period. It is shown that during wave-movement this centre is elevated 

 above the level that would be occupied by the molecule in repose, a 

 height due to the vis viva of the molecule. 



The profile of any wave-stratum is a trochoid, the length of which is 

 the distance from hollow to hollow or from crest to crest, and the height 

 is the diameter of the orbit of the molecule belonging to that stratum. 

 The highest possible wave is that where the trochoid becomes the 

 cycloid, or where the length of the wave is equal to the circumference of 

 the orbit. No trochoid of greater height is physically possible, as such 

 a curve must have a looped crest, where the liquid molecules would have 

 to cross the paths of each other, producing broken water. 



The velocity and period of a wave, and the angular and actual velo- 

 cities of the liquid molecules, are deduced in terms of the length of the 

 wave. 



The general results of the investigation are shown by the following 

 formulae, in which the symbols employed are : 



L= length of wave from crest to crest. 



v=. velocity, or distance apparently traversed by the wave in a given 

 unit of time. 



T = the period, or time occupied by the passage of the whole wave. 



g=. gravity (32 feet per second). 



R= radius of the orbit of a molecule at 



H = height measured from bottom, and 



r= radius at 



h= height. 



x= horizontal, and 



y= vertical ordinate of molecule in stratum at height h and at time , 

 from the epoch when the molecule is at its lowest point, or when 

 tf=0. 

 Then 



=r sin zt 



y= r cos Zt, 



v 



origin being the centre of orbit, 



/2*L 



\/ 



" V g 



9 = angular velocity = \ / -$- . 

 v \ Lt 



