344 REPORT — 1844. 



rfD and D d' be divided into the same number of equal parts. Let there be 

 drawn through each division of the circles horizontal lines, and through each 

 division of tlie wave lengths let there be drawn perpendiculars, meeting suc- 

 cessively the horizontal lines in 9, 8, 7, 6, 5, 4, 3, 2, 1, — these will be points 

 in the curve of versed sines, that is of the (approximate) form of the wave. If, 

 therefore, we conceive the wave-form to move horizontally and uniformly 

 along the line dD d', and at the same time a particle of water on the surface 

 to rise successively to the heights 1, 2, 3, 4, 5, and fall vertically to 6, 7, 8, 9, 

 on the diameters cd and c' d', then the place of the particle will always coin- 

 cide with the wave curve. 



This is the same form (only wholly positive) which Laplace assigns to the 

 tide wave in the ' Mecanique Celeste,' torn. iii. liv. iv. chap. iii. Art. 17. " Con- 

 cevons un cercle vertical, dont la circonference en partant du point le plus bas, 

 expriment les temps ecoules depuis la basse ; les sinus verses de ces arcs, 

 seront les hauteurs de la mer, qui coiTespondent a ces temps." Or as he says 

 elsewhere, " Ainsi, la mer en s'elevant, baigne en temps egal, des arcs egaux 

 de cette circonference." So if we imagine a circular disc placed vertically so 

 as to touch the surface of the water in repose, the passing wave will in suc- 

 cessive equal times cover equal successive arcs of the circumference. 



The wave is of this form when its height is small, and the deviation in- 

 creases with the increase of height. 



Vertical Motion of each Particle. — No more then is necessary to the exhibi- 

 tion of the wave curve than that every particle of the surface of the water 

 should be made to rise and fall successively, according to the increase and 

 decrease of the versed sines of the circle of height. Let us follow the mo- 

 tion of a single particle. Draw c' d' a vertical diameter of the wave circle, 

 suppose C efg h c' the successive places of the wave crest in successive equal 

 intervals of time, 1, 2, 3, 4, 5, 6, 7, 8, 9, successive versed sines on cc? and 

 c' d' of equal arcs of the wave circle. When the wave centre is at C, the 

 particle is at d'. When the wave centre is at e, the particle has risen to 1. 

 When the wave centre has reached/, the water particle has risen to 2. When 

 the wave has advanced to g li c', &c., the water particle has risen to 3, 4, 5, 

 &-C. ; and if every successive particle along the surface be conceived to per- 

 form successively a similar series of vertical motions, the surikce of the water 

 will present to the eye the visible moving wave form. Such is the simplest 

 geometrical mode of exhibiting to the eye and of conceiving wave motion of 

 the first order ; it approximately represents the form of a wave of the first 

 order whose height is small. 



Horizontal Motion of each Particle. — This mode of representing the wave 

 motion is inaccurate, in so far as it does not take account of the horizontal 

 motion, which must of necessity accompany the vertical elevation of the water. 

 Water being an inelastic fluid, any vertical column of the liquid can only 

 have its length increased by a diminution of its horizontal dimension. It is 

 necessary, therefore, to represent or conceive this horizontal motion as well 

 as the vertical motion. 



The horizontal range of motion of the wave is necessarily determined by 

 the volume of the wave. The water which forms the wave is added to the 

 given volume in which the wave is formed, at its posterior extremity, and 

 thence displaces a new volume of water which goes to displace the volume 

 of the wave in the next portion of the channel. Thus the volume of water 

 which occupied the space A' B' b d before the transit of the wave (see Plate 

 LII. fig. 4), occupies only the length AB b d during the wave transit, and it 

 now consists of the rectangle AB bd, together with the volume of the wave 

 A C d, which volume is equal to the volume A B B' A' by which it is re- 



