Dr. Rankine on the Exact Form and Motion of Waves. 421 



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 that revolves in the same period 

 with the particle of liquid. 



Proposition II. — Let another surface of uniform pressure be 

 conceived to exist indefinitely near to the first surface : then if the 

 first surface is a surface of continuity (that is, a surface always 

 traversing identical particles), so also is the second surface. 



(Those surfaces contain between them a continuous layer of 

 liquid.) 



Corollary. — The surfaces of uniform pressure are identical with 

 surfaces of continuity throughout the whole mass of liquid. 



Proposition III. — The profile of the lower surface of the layer 

 referred to in Prop. II. is a trochoid generated by a rolling circle 

 of the same radius with that which generates the upper surface ; 

 and the tracing-arm of the second trochoid is shorter than that 

 of the first trochoid by a quantity bearing the same proportion 

 to the depth of the centre of the second rolling circle below the 

 centre of the first rolling circle, which the tracing-arm of the 

 first rolling circle bears to the radius of that circle. 



Corollaries. — The profiles of the surfaces of uniform pressure 

 and of continuity form an indefinite series of trochoids, described 

 by equal rolling circles, rolling with equal speed below an inde- 

 finite series of horizontal straight lines. 



The tracing-arms of those circles (each of which arms is the 

 radius of the circular orbits of the particles contained in the 

 trochoidal surface which it traces) diminish in geometrical pro- 

 gression with a uniform increase of the vertical depth at which 

 the centre of the rolling circle is situated. 



The preceding propositions agree with the existing theory, 

 except that they are more comprehensive, being applicable to large 

 as well as to small displacements. 



The following proposition is new. 



Proposition IV. — The centres of the orbits of the particles in 

 a given surface of equal pressure stand at a higher level than the 

 same particles do when the liquid is still, by a height which is 

 a third proportional to the diameter of the rolling circle and the 

 length of the tracing-arm (or radius of the orbits of the particles), 

 and which is equal to the height due to the velocity of revolution 

 of the particles. 



Corollaries. — The mechanical energy of a wave is half actual 

 and half potential — half being due to motion, and half to eleva- 

 tion. 



The crests of the waves rise higher alove the level of still 

 water than their hollows fall below it; and the difference 

 between the elevation of the crest and the depression of the 



