1862.] 277 



The preceding propositions agree with the existing theory, except 

 that they are more comprehensive, being applicable to large as well 

 as small displacements. The following proposition is entirely 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 pro- 

 portional 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 

 crests and the depression of the hollows is double of the quantity 

 mentioned in Proposition IV. 



The hydrostatic pressure at each individual particle during the 

 wave-motion is the same as if the liquid were still. 



Friction between a Wave and a Wave-shaped Solid. 

 In an Appendix is given the investigation of the problem, to find 

 approximately the amount of the pressure required to overcome the 

 friction between a trochoidal wave-surface and a wave-shaped solid in 

 contact with it. The application 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 useful of the 

 formulae arrived at. Let w be the heaviness of the liquid ; f y the 

 coefficient of friction ; ff, gravity ; v, the velocity of advance of the 

 solid ; L, its length, being that of a wave ; #, the breadth of the 

 surface of contact of the solid and liquid ; /3, the greatest angle of 

 obliquity of that surface to the direction of advance ; P, the force 

 required to overcome the friction ; then 



sin 4 /3). 



In ordinary cases the value of/ for water sliding over painted iron is 

 about -0036. The quantity Lgr(l + 4 sin 2 /3 + sin 4 /3) is what has been 

 : called the "augmented surface." In practice, sin 4 /3 may in general 

 be neglected on account of its smallness. 



