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



which is the period of a revolving pendulum, or the time of a 

 double oscillation of a simple pendulum whose height is R. 

 Compare (10). 



7. The dependence of a wave's period on its length alone, not on 

 its height. — This is a corollary from the preceding. The period 

 varies as the square root of the length, and is the same for all 

 subwaves as for the surface-wave, the length being the same for 

 all. The height, within certain limits, is independent of the 

 length, as appears in the apparatus, and as may be inferred from 

 the formulae given further on. It depends on the centrifugal 

 force of the particle, and this ultimately on the external forces 

 generating it. 



8. The varying direction and intensity of the resultant force 

 acting at each instant on a given particle in a wave. — The compo- 

 nent forces are two — the particle's gravity and its centrifugal 

 force. The former is represented by the vertical radius of the 

 large circle, the latter by the radius vector of the revolving par- 

 ticle ; their resultant, then, is represented by the third side of 

 the triangle of forces, or the side formed by the wire pendulum. 

 This resultant must be always normal to the wave-surface, as the 

 wire pendulum is seen to be always at right angles to the elastic 

 wire representing that surface. 



9. The condition of a wave's rupture at the crest. — When 

 the centrifugal force becomes equal to gravity (or the radius of 

 the orbit to that of the large circle), the resultant force for a 

 particle at the highest point of its orbit or crest of the wave must 

 be zero, and the particle consequently fly from its orbit, or the 

 crest break in foam. 



10. The trochoidal form of the wave-curve. — The point of 

 suspension of the pendulum, that is, the upper extremity of the 

 vertical radius of the large circle, may be regarded as the in- 

 stantaneous centre about which an element of the wave-curve at 

 the point of normality of the pendulum is described. Conse- 

 quently, if this circle be rolled under a horizontal straight line, 

 a point within it distant half the height of a wave from the centre 

 will trace the wave-profile, which therefore is a trochoid. The 

 rolling circle is the same for all wave-profiles down to still water, 

 the lengths of the tracing-arm only differing. The circumference 

 of this circle equals, of course, the wave's length. 



11. The greater sharpness of the crests than of the troughs 

 of waves. — This follows from the preceding, and is shown in 

 the relative positions of the crank-pins — nearer together at the 

 crests, further apart in the troughs. The trochoids become 

 necessarily sharper at the upper bend, and less so at the lower, 

 as the tracing-arm approaches to an equality with the radius of 

 the rolling circle ; until, when that equality occurs, the trochoid 



