518 Mr. A. J. Robertson on the Positive Wave of Translation. 



motion in equal space and time. When the surface rises to 

 E it comes to rest. The downward force continuing to act, 

 becomes an accelerating force ; at T the velocity is a maxi- 

 mum; again it is retarded ; and when the surface has sunk to 

 B, it is again at rest. It is to be borne in mind, that whilst 

 the uppermost particles in this column have been moving 

 through a vertical distance A to E, the lowest particles have 

 not been moving at all ; and the intermediate particles have 

 had an intermediate movement, as in the case of the single 

 column. Also taking the columns sufficiently small, the hori- 

 zontal motion of the particles in each may be disregarded, 

 their total effect being the horizontal movement of the whole 

 column. 



From equation (4.) we have seen that the time of rising 

 from A to E by a single column, urged by a force varying as 



above, is proportional to \f — — ; — ^- is the absolute force 



r r V 3g a + k 



on the column at a distance unity from QN. But the force 

 which urges the column of the wave cannot be the same as 

 this, as it is modified by the horizontal motion of the rest of 

 the columns. It must nevertheless vary in the same propor- 

 tion. Let the force be ^ then 



r a + tf 



the value of c remains to be determined. 



Every column between the commencement and crest of the 

 wave is becoming continually narrower and higher ; therefore, 

 although the inclination of the surface is greatest at R, the 

 forward motion is greatest at E. But at R the velocity of the 

 column upwards is at its maximum; the rate^ therefore, at 

 which it shrinks is greatest. After the crest of the wave has 

 passed, the horizontal force becomes a retarding force, and 

 destroying the motion in the same time as it had communicated 

 it, the end of the wave finds the column at rest*. What, 

 then, is the quantity of this motion ? 



* The peculiarity of the wave of translation appears to be this — the 

 horizontal and vertical movements commence and end together. Suppose 

 that by some means a backward horizontal motion had been communicated 

 to the column AG, such that by the time the surface has risen to R it is 

 destroyed. Whilst the path of the particles in space is altered by this 

 supposition, the relation between the vertical and horizontal movements 

 remains the same. The shrinking of the columns would only be towards 

 R in place of towards A : at R the upward movement would still be a maxi- 

 mum. The horizontal motion would then be reversed, acquire a maximum 

 at E, be destroyed at T, and reversed again. The motion being therefore 

 alternately backwards and forwards, the particles oscillate about a fixed 



