276 [Nov. 27, 



Having communicated some of the leading principles of that in* 

 vestigation to Mr. William Froude in April 1862, the author was 

 informed by that gentleman that he had arrived independently at 

 similar results by a similar process, although he had not published 

 them. 



The following is a summary of the leading results demonstrated in 

 the paper. 



Proposition I. -In a mass of gravitating liquid whose particles 

 revolve uniformly in vertical circles, a wavy surface of trochoidal 

 profile fulfils the conditions of uniformity of pressure ; 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 that revolves in the same period with the particles of 

 liquid. 



Proposition II. Let another surface of uniform pressure be con- 

 ceived 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 re- 

 ferred to in Proposition 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 indefinite series of 

 horizontal straight lines. 



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

 the circular orbits of the particles contained in the trochoidal surface 

 which it traces) diminish in geometrical progression with a uniform 

 increase of the vertical depth at which the centre of the rolling circle 

 is situated. 



