4 N. M. FENNEMAWN 
The path generated by any point within the circle is a trochoid. 
This line will be sharply curved or broadly curved, approaching 
straightness, according as the point which generates it is chosen 
near the circumference or near the center. The distance from 
the center to the generating point is called the tracing arm. 
When the point is at the circumference —that is, when the trac- 
ing arm equals the radius of the rolling circle—the curve is 
cusped at the top and is the common cycloid (see Fig.8). This 
is the steepest and shortest form which a true wave can have. 
If the tracing arm be longer that the radius of the moving circle 
the curve is looped instead of cusped (/zg. 4). The failure of 
the water surface to assume these looped forms results in 
breakes. 
Steepness dependent on differential movement.— lf the same series 
of particles in their orbits be represented in several diagrams, 
assuming for each diagram a different amount of differential 
movement, the wave will be found to be long when the differen- 
tial movement is small, and short when the differential movement 
is large (compare Figs. and 2): If the size of the orbits be 
increased while the distance between the particles remains the 
same, and at the sametime the differential movement continues 
to be a certain arc of the orbit, the wave-length remains the 
same, but its height and steepness increase (compare Figs. 1 
and 3). If the size of thc orbit be increased and the differential 
movement remain the same in absolute amount, instead of the 
same in arc, the shape of the wave will be preserved and its 
dimensions increased with the dimensions of the orbit (compare 
Figs. 2 and 3). If the differential movement exceeds a certain 
limit the curve will loop (see Fig. 4). This condition corre- 
sponds to that of breaking waves as noted above. 
Movement of particles below the surface —Vf a series of equidis- 
tant particles be considered which lie in a vertical line in still 
water, the movement of the topmost or surface particle is repre- 
sented by any one of the orbits considered above. That of the 
second one is similar in every way except in size of orbit and 
hence invelocity. Its orbit issmaller and described in the same 
