1874.] Uniform Wave of Oscillation. 351 



capable of ready geometrical demonstration, that all molecules which in 

 repose would be at the same level, move in equal and similar trajectories, 

 but that each molecule towards the one hand, as towards the right, is by 

 a certain interval of time in advance of the contiguous molecule on the 

 left. It is also taken as a necessary condition of the wave-movement 

 that the excursions of the molecules are periodic, and effected in closed 

 orbits returning into themselves. 



With these general postulates, the author proceeds to investigate first 

 the conditions necessary to maintain continuity of the liquid, or the con- 

 stancy of the vertical sectional area of an elementarj^ portion of the 

 liquid, in all parts of its orbit. He then traces the operation on such an 

 element of the forces to which it is subjected, these forces being gravity, 

 or the weight of the element itself, and the pressure directed on it by 

 the surrounding liquid. 



The liquid in repose is supposed to be divided into numerous hori- 

 zontal strata, each stratum forming an undulating film when the wave- 

 movement is established. The length of any such stratum is supposed 

 to be divided into numerous portions, the width of each of which is the 

 distance apparently traversed by the wave in a very short interval of 

 time. By taking the depth of a stratum, and the interval of time which 

 determines the width of one of its divisions, such that the element of 

 liquid may be considered a parallelogram of constant area, the several 

 differential equations expressing the continuity of the liquid and the 

 effect of the forces on the element are developed in an integrable form. 



The parallelogram representing the liquid element is determined in its 

 form and position by the position of the points at its four angles. 



One of those points, namely that at the lower left-hand angle, is 

 assumed to move in a path the horizontal and vertical coordinates of 

 which, x and y, are referred to an origin situated at a height A mea- 

 sured from the bottom of the liquid, and the position of the point in its 

 path is taken at a time t reckoned from the epoch when the point was 

 vertically under its origin. The point at the lower right-hand angle of 

 the parallelogram is referred to an origin on the same level with the 

 former, but separated horizontally from it by a space, v&t, where v is the 

 apparent velocity of the wave, and A* is the short interval of time by 

 which the one point is in advance of the other in its trajectory. The 

 coordinates x and y of the first point being functions of h and t, those 

 of the other point are the same functions of h and t-\- A. The upper left- 

 hand point being referred to an origin which is at a height AA above the 

 level of the former origins, but being taken as contemporaneous in its 

 movement with the point below it, its coordinates are functions of 

 &-f A7t and t; and in like manner the coordinates of the upper right- 

 hand point are functions of A + AA and t+bt. 



As it does not a priori appear that the origin of the upper point must 

 be vertically above that of the lower (though in the course of the inves- 



