Mr. A . J. Robertson on the Positive Wave of Translation. 513 



all in the same direction, and the extent of the transference is 

 equal throughout the whole depth. Hence this wave may be 

 descriptively designated the Great Primary Wave of Trans- 

 lation. The motion of translation commences when the an- 

 terior surface of the wave is vertically over a given series of 

 particles; it increases in velocity until the crest of the wave 

 has come to be vertically above them, and from this moment 

 the motion of translation is retarded, and the particles are left 

 in a condition of perfect rest at the instant when the posterior 

 surface of the wave has terminated its transit through the 

 vertical plane in which they lie. This phenomenon has been 

 verified up to a depth of five feet." 



Lemma I. Let the vessel AH contain a quantity of water 

 the surface of which is any 

 curve AEB. 



Let PQ pq be vertical ele- 

 mentary columns of the water, 

 and let CD be a horizontal line 

 bisecting the difference of level 

 of A and B. 



~Lety=j[x) be the equation 

 to the curve ; then the acce- 

 lerating force in a horizontal 



p 





S *\jE r 





R \s 



P 





a 





dy m 

 dx* 



for 



direction on every particle in the column PQ is g 



~ expresses the ratio of the difference of two consecutive or- 



dinates to the distance between them — the moving force to 

 the mass moved. Every particle in the column PQ is like- 

 wise pressed upwards by a force represented by AC — PR. 

 They are pressed downwards by a force PS= PR -j- RS 5 there- 

 fore downward pressure 



= PR + RS-(AC-PR) = 2PR, for RS---AC. 



In the column pq the pressure is upwards and = 2pr. 



Therefore generally, calling the upward pressure negative 

 and the downward positive, the vertical pressure on any column 

 equals the weight of a column of the same section, the length 

 of which is twice the distance of its summit above the line CD. 



Hence at E there is no vertical pressure. 



Lemma II. The oscillation of an exceedingly thin vertical 

 column of fluid, urged by a force varying as the distance of 

 the top of the column from a given point, is isochronous. 



Let AgcB be a column of water. 



Let there be a thin plate between this column and the 



Phil Mag. S. 3» No. 253. SuppL Vol. 37. 2 L 



