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



water in the vessel APQB, the plate 

 being capable of horizontal motion. 



Let DC be the column when raised 

 above the level of repose, and let the 

 water in the vessel be supposed to have 

 no influence on the water in the column 

 DC,except exerting a pressure upon it; 

 because if this water were to be put in 

 motion by the motion of the column 

 .DC, the conditions would be materi- 

 ally altered from those assigned in the 

 enunciation. Q 



The unbalanced pressure on the column DC == weight of 

 column DG. 



Let area of column Ac =/3 "^ AD = x. 



And the thickness == unity. 



AB=a+k 

 Let the distance from which motion 



begins AN=& 



In order that the surface T)d may fall, the plate must move 

 horizontally from AB, the area of the column being constant; 

 and the movement of the top and bottom of the plate will be 

 the same, because the pressure on every part of it is the same. 

 Although the surface Da? falls, the particles at BC are, as 

 regards vertical motion, at rest; and the vertical movement 

 of any intermediate row of particles is proportional to its di- 

 stance from B. 



In the same manner the surface dC moves horizontally 

 whilst the surface DB is at rest, and the horizontal movement 

 of any intermediate row of particles is proportional to its di^ 

 stance from the line DB. 



Now the inertia of a column DB, the particles of which 

 have a motion varying from a given quantity 7 at the top to 

 nothing at the bottom, is one- third of that of a similar column 

 every particle of which is moved through 7, and / being the 



length, inertia = 1 — . 



The inertia, as regards horizontal movement likewise, is 



- of what it would be if every particle had the same motion 

 3 



as the surface dC; and the total inertia is the sum of the 



two. 



Let the surface T)d fall through space Ax, and let Aj/ be 



the corresponding horizontal movement of dC, then 



