Polytechnic Association Proceedings. 777 



below the centers of the orbits, the upper one of these being the 

 surface line, the lower one a line of particles one-ninth of a wave's 

 length below. The upright wires represent lines of particles which 

 at rest would be vertical. Every point in these moving lines des- 

 cribes its own distinct orbit. The spaces between the wires show 

 the varying distortions of sections of water originally rectangular. 

 The radius of the large circle is made such that its ratio to the 

 radius of a particle's orbit is equal to the ratio of gravity, or the 

 weight of the particle, to its centrifugal force; or, putting R and 

 r for the radii respectively, and t for the time of revolution, we 



4 7T^ T I Tf 



make R : r :: g \ — —] hence, 1 = 2 Tri/_, which is the period 



t y g 



of the revolving pendulum whose height is R. Such a pendulum, 

 then, will keep time with the wave. 



The force acting on a particle being the resultant of its weight 

 and centrifugal force, if the latter be represented, in intensity and 

 direction by the crank-arm or radius vector of the particle, the 

 former, or gravity, Avill be represented, similarly, by the vertical 

 radius of the large circle, and the resultant of the two by the third 

 side of the triangle, or a line drawn from the top of this radius to 

 the particle. The wire pendulum represents this resultant, and, 

 like it, is always normal to the wave surface. 



The measure of the resultant force being the distance on the wire 

 pendulum from its point of suspension to the wave surface, it is 

 seen that when the particle is at the top of its orbit, the acting force 

 is its weight minus its centrifiigal force; when, then, the centri- 

 fugal force equals the weight, as in high short waves, the resultant 

 becomes zero, and the particle, no longer held by gravity, flies 

 from the crest in foam. 



Since the wire-pendulum is always normal to the wave-curve, its 

 entre of oscillation is the instantaneous centre about which may be 

 described an element of the curve at the point of normalcy. 

 Hence if this circle be rolled along under a horizontal straight line, 

 a point within it, at a distance from its center equal to half the 

 height of a wave, will describe a trochoid, which is the wave pro- 

 file. Consequently, the circumference of the rolling circle is equal 

 to the length of a wave; and the period of a wave is that of a 

 revolving pendulum whose height is the radius of the same circle. 



If the describing point be taken in the circumference of the roll- 

 ing circle, the curve is an inverted cycloid. The cycloidal cusp, 



