Mr. A. J. Robertson on the Positive Wave of Translation. 517 



ment which must hike place in all the water behind, in order 

 that the pressure upon the column may be sustained. When 

 the column AG rises, it necessarily becomes thinner; whilst 

 pg, moving forwards to supply the place which would other- 

 wise be left vacant^ exerts a pressure on AG, it is itself pressed 

 by a higher column ; and as it can move forward only so far 

 as the horizontal diminution of AG permits, it also rises — and 

 so with every column behind. The further it is from A, the 

 greater evidently must be its horizontal motion for a given 

 upward movement of column AG, because it must pass through 

 a space equal to the sum of all the compressions in front of it, 

 and this is true of any column between A and the crest E of 

 the wave. 



We have seen before, in the case of the water contained in 

 the vessel, that the accelerating force forwards at any point is 



d\i 

 proportional to ~ — the tangent of the inclination of the sur- 

 face at the point ; also that there is an upward pressure, which 

 varies as the distance of the column below the line QN. At 

 the commencement A of the wave, the conditions have been 

 shown to be the same for the instant as if the water were in a 

 vessel of which the end is solid — the pressure is wholly up^ 

 wards. As we recede from A, the motion forwards requires 

 to be greater and greater ; and the slope of the surface in- 

 creases as well, until about the point R it attains its maximum 

 value. On the other hand, the upward pressure becomes less 

 and less, until at the point R there is no vertical pressure at all. 

 The forward motion then arises only from the compression of 

 the columns, and is limited by it; but the rising of the columns 

 in succession evidently produces that change in the position 

 of the wave-form which gives the water the appearance of 

 actual forward motion ; and if the wave be divided into an 

 exceedingly great number of vertical columns, each of these 

 columns represents a stage that every one of the others either 

 has passed through or has yet to pass through. Since either 

 of the motions of a body urged by one force in a vertical, and 

 by another in a horizontal direction, is independent of the 

 other, the vertical motion of these columns will be the same, 

 and may be examined as if they had no horizontal movement. 

 It appears, then, that the column AG is urged upwards by 

 a force proportional to k, and diminishing as the distance of 

 its summit from QN diminishes, until at R it is nothing. It 

 in consequence acquires a certain velocity, by virtue of which 

 it continues to rise and get thinner. Now, however, there is 

 a retarding force acting downwards, which, as it varies in the 

 same proportion as the accelerating force does, destroys the 



