6 REPORT—1862. 
horizontal axis; but as the theory of such wayes was foreign to the subject of the 
book, he deferred until now the publication of the investigation on which that 
statement was founded. ae 
Having communicated some of the leading principles of that investigation to 
My. William Froude in April 1862, the author was informed by that gentleman that 
he had arrived independently at similar results by a similar pees although he had 
not published them. The introduction of Proposition I. between Propositions I. 
and III. is due to a suggestion by Mr. Froude. 
The following is a summary of the leading results demonstrated in the paper :— 
Proposition 1.—In a mass of gravitating liquid whose particles revolve uniformly 
in vertical circles, a wavy surface of trochoidal profile fulfils the conditions of uni- 
formity 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 eaters of liquid. 
‘Proposition 11.—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 con- 
tinuity (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 con- 
tinuity throughout the whole mass of liquid. 
Proposition I1I.—The profile of the lower surface of the layer referred to in Pro- 
position 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 frochoid 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 arms 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. 
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 is new as an exact proposition, although partly anticipated by the 
approximative researches of Mr. Stokes :— 
PropositionTV.—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 elevation. 
The crests of the waves rise higher above 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 hollow is double of the quantity mentioned in Proposition II. 
The hydrostatic pressure at each’ individual particle during the waye-motion is 
the same as if the liquid were still.’ * 
In an Appendix to the paper is given’ the investigation of the problem, to find 
approximately the amount of the pressure required to overcome the friction between 
a trochoidal waye-surface and a wave-shaped solid in contact with it. The appli- 
cation of the result of this investigation to the resistance of ships was explained 
in a paper read to the British Association in 1861, and published in various 
engineering journals in October of that year. The following is the most conve- 
nient of the formule arrived at:—Let w be the heaviness of the liquid; f the 
coefficient of friction; gy gravity; v the velocity of advance of the solid; L its 
length, being that of a wave; 2 the breadth of the surface of contact of the solid and 
liquid; 8 the greatest angle of obliquity of that surface to the direction of advance 
a 
