THE TWO GREAT SOLITARY WAVES OF RUSSELL. 329 



observations, that a thiti solid plane transverse to the direction of transmission, and so poised 

 as to jloat in that position does not sensibly interfere with the motion of translation or of 

 transmission.^'' 



From this statement it would appear that we may safely assume, as an experimental fact, 

 the second principle which I have proposed to assume as the basis of calculations. The 

 observations required to be made in establishing it are such as admitted of very accurate 

 verification ; and seem also to have been made with care, and therefore the principle must l)e 

 either accurately true or very nearly so. By reference to the Report itself the reader will find 

 that this property of the solitary wave is not shared by any of the other three species of waves, 

 and is therefore very proper to serve as a distinctive assumption to sift this species from the 

 general equations of fluid motion. The investigations which follow will therefore contain the 

 Mathematical Theory of Waves of the First Species, i.e. of the Positive and Negative Solitary IVaves. 



PROBLEM. 



A QUANTITY of incompressible fluid is in a state of repose in a straight horizontal canal, the 

 sides of which are vertical and parallel, and the bottom horizontal. A single wave is generated 

 by pushing in one end of the canal in a proper manner : to determine the subsequent motion of 

 the fluid, on the two hypotheses before mentioned, viz. 



1st. That the velocity of transmission of the wave is uniform. 



And 2nd. That the horizontal velocity of every particle, in a transversal section of the 

 canal, is the same. 



Let a horizontal line drawn along the bottom of the canal, parallel to the sides, be taken 

 for the axis of a; ; let the axis of y be vertical. 



h = equilibrium depth of the fluid ; 



k = the depth from the top of a wave to the bottom of the fluid ; 



c = the velocity of transmission of the wave. 



As the motion of each particle is manifestly in a vertical plane, it will not be necessary to 

 take account of the breadth of the canal, nor of the third co-ordinate of any particle ; let 

 therefore .ry be the co-ordinates and ttv the velocities of any particle at the time t ; and suppose 

 p the pressure of the fluid at the same point ; the density of the fluid being taken as unity. 



Then by our second hypothesis m is a function of .i- and not of y; consequently the equations 

 of motion are in this case, 



d^p = - diU - nd^u (1), 



dp=—g-d,v-ud^v-vd^v (2); 



and the equation of continuity is, 



= d,u + d^v (;;), 



and our first hypothesis gives, 



= d,u + cdji (4). 



■From those four equations we are to obtain our results. 



Vol.. VIH. Pakt IIL Uu 



