778 Transactions of the American Institute. 



then, is the limit of sharpness of a wave's crest, the radius of the 

 orbit being, in that case, equal to that of the rolling circle, or in 

 other words, the particle's centrifugal force equal to its weight. 

 All possible wave protiles, therefore, are trochoids of curvatures 

 ranging between the limits of the cycloid on the one hand, and the 

 straight line on the other, or between the sharp crest breaking in 

 foam and the level of still water. The greater sharpness of the 

 crests than of the troughs, with its cause, is conspicuous in the appa- 

 ratus. 



It is seen also that the crests rise higher above the level of still 

 water than the troughs fall below it. The difference is equal to 

 twice the height due to the orbital velocity of the particle, or is a 

 third proportional to the radius of the rolling circle and the radius 

 of the particle's orljit; that is, putting R and r for these radii 

 respectively, and D for the difference in question. 



^ = ^- 



The rolling circle is the same for all wave profiles (or surfaces 

 of equal pressure) down to still water, the tracing arm, or orbit- 

 radius, only becoming shorter in a geometrical ratio, with increase 

 of depth. The rate of shortening is approximately one-half for 

 each additional depth equal to one-ninth of a wave's length; or 

 more exactly, putting r and r' for the radii respectively of a sur- 

 face orbit and of one whose middle depth is /,-, it is 



k_ 

 r' = r e ^t 

 R Ijeing the radius of the rolling circle, and e the base of the Nape- 

 rian logarithms. 



The distance of the crank-axes above the horizontal lines on the 

 background corresponds to the distance of the orbit-centers of cor- 

 responding particles in wave motion above the position of the same 

 particles when at rest; this distance is a third proportional to the 

 diameter of the rolling circle and the radius of the particle's orbit, 

 or is equal to the height from which a body must fall to acquire 

 the orbital velocity of the particle, or is equal to the area of the 

 orbit divided by the length of the wave; that is, putting h for this 

 distance, I for the wave's length, v for the orbital velocity, and the 

 other symbols as before, 



A = 



2R~ 2 ge 2g I 



