On C. S. Lyman's new form of Wave-apparatus. 19 



passes into the cycloid, which has sharp cusps. The cusp of the 

 inverted cycloid, then, is the limit of sharpness of a wave's crest. 

 The equality above named is equivalent to that of the centrifugal 

 force of a particle with its gravity (9). When the latter condi- 

 tion occurs, the wave-curve is cycloidal, and only then. 



12. The limits of possible curvature of waves. — That curvature 

 must always lie between the cycloid at the one extreme, and the 

 straight line at the other, embracing trochoids of every possible 

 variety. 



13. The greater elevation of the crests above the level of still 

 water, than depression of the troughs below it. — The difference be- 

 tween this elevation and depression is equal to twice the height 

 due to the orbital velocity of the particles — that is, to twice the 

 height from which a body must fall to acquire that velocity — or 

 is a third proportional to the radius of the rolling circle and that 

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



respectively, v for the orbital velocity ( = ) , and D for the 



difference in question, 



-p. _ r 2 _ v* 2 



■ R~7' 



When r equals R, then D = r, or half the height of the wave. 



14. The elevation of the centres of the orbits of particles above 

 the positions of the same particles at rest. — This is shown in the 

 distance of the axes above the corresponding lines on the back- 

 ground. These lines show the positions of lines of particles at 

 rest, which in motion form the wave-profiles represented by the 

 transverse wires. The elevation in question is equal to the height 

 due to the particle's orbital velocity, or is a third proportional to 

 the diameter of the rolling circle and the radius of the orbit, or 

 is equal to the area of the orbit divided by the length of the wave ; 

 that is, putting H for this elevation, / for the wave's length, and 

 the other symbols as before, 



^ r" 2 _ v 2 _ 7rr 2 

 2R~%~T* 



r 

 When r equals R, H = -, or one-fourth the height of the wave. 



To this elevation is due one-half the mechanical energy of a wave, 

 the other half to the motion of its particles. That energy is, in 

 other words, half potential, half actual. 



15. The decreasing diameter of the orbits with depth. — This is 

 seen in the shorter crank-arms below, and the decreasing ampli- 

 tude of sway of the upright elastic wires down to their points of 

 rest, which mark the depth of still water. The decrease of the 

 orbits in diameter takes place in a geometrical ratio, and is ap- 



C2 



