and their Action on Floating Bodies. 83 



ject, placed it on too sound a basis for further controversy. 

 They have shown that, amongst other properties, 



4. The orbit of each particle of water in wave-motion is 

 an ellipse ; the form of which depends in a known law on 

 the depth of water, so that in the ocean, the depth of which 

 is approximately infinite, the orbit is circular.* 



5. That the wave-surface is trochoidal in shape : ("Rankine, 

 Manual of Applied Mechanics' first published in 1858: "On 

 exact form and motion of waves," Trans. Roy. Soc. 1862) 

 such trochoidal profile being generated by rolling on the 

 under side of a horizontal straight line, a circle whose 

 radius is equal to the height of a conical pendulum, which 

 revolves in the same period with the particles of liquid. 



6. The hydrostatic pressure at each particle is the same 

 as if the liquid were still. From these propositions, viz., 4, 

 5, and 6, which, for the sake of distinction, may be called 

 Eankine's laws, there follow that — 



7. If h be the height of a wave in deep water, its length 

 is never less than tt h. 



. 8. The undulations of waves are performed in the same 

 time as the oscillations of a pendulum whose length is equal 

 to the breadth of the wave, or to the distance between two 

 neighbouring cavities or eminences (see Art. Hydrodynamics 

 JEnc. Brit. 8tli ed. p. 162.) Now the oscillations of 

 pendulums being as the square roots of their radii, it hence 

 follows that 



9. The periods of waves are as the square roots of their 

 lengths."!" 



* In the propositions which follow, ocean-waves only are referred to. As 

 in this paper frequent reference is made to previous passages, it has been 

 found necessary to number them throughout. 



t The length of the seconds pendulum at the Pole is 39-218 inches; at 

 the Equator 39-018, whence the mean length = 39-118. Now the length of 

 a pendulum whose period of oscillation is <^ 89-118 <2 in inches; conse- 

 quently if the length of a wave = 6 in feet, and v be its velocity in feet 

 per second, we shall have b= '3-2Q fi and v = 3-26 < : also < = '^',3'26" 

 from which equations all problems relating to cycloidal or breaking waves 

 (which are thfse to which Smeaton's dogma more immediately referred) 

 may be solved. If foi- b we substitute its value, 3-141C/«, then t = 

 \/ 0.963 //,. Or finally— 

 6 = 319 16 A. 



t = 0-31 Ct3 //( where t = time the wave transverses space = its cwn length; 

 V = 1-0116 /h. 



By which it appears that the velocity of a cycloidal wave per second ^is 

 very nearly = square root of its height in feet 



G 2 



