Mr. J. M c Cowart on the Solitary Wave. 53 



and from (7) and (8) , or directly from (33) or (34) , the com- 

 ponents j and f of the velocity a will be given by 



• 1 + cos mz cosh mat „ e N 



t=-uma, r - i K2 , . . . (oo) 



(cos mz + cosh m/c') 



£ TT sin ww' sinh mx- /nn , 



£== \Jma-. — — r- r 7To, . . . U>b) 



(cos mz + cosh m^ y v 



and 



.-. a- = U??ia/(cos??i^ + cosh??i^ / ). . . . (37) 



From (37) and (35) we see that the whole velocity and 

 its horizontal component at any instant are nearly constant 

 for all particles in the same vertical line, while the vertical 

 component is, by (36), roughly proportional to the distance 

 from the bottom. Further, from (37) we see that at the end 

 of the wave, as we have defined it in Section 5, the velocity is 

 only about 0*16 of the velocity in the centre of the wave, and 

 that it decreases with extreme rapidity as we go further from 

 the centre. 



If be the inclination to the axis of x of the path of a 

 particle initially at the distance z from the bottom, then bv 

 (35) and (36) 



t a __ y it — sin mz ' sinn mx ' C3«\ 



^ 1 + cos mz' cosh mx' ' • v >> 



therefore, since initially x' = co , each particle begins to move 

 forward from rest at an inclination, 6=mz, proportional to 

 its distance from the bottom and inversely proportional to 

 the length of the wave ; its velocity goes on increasing till 

 x + %=TJt, when it moves horizontally with its maximum 

 velocity, and it finally returns to rest at an inclination 

 0=—mz equal and opposite to that with which it started. 

 We proceed, however, to seek the actual paths described by 

 the particles. 



8. The Paths of the Particles. 



If we integrate (36) we get 



arinW 



cos mz + cosh mar v ' 



This immediate integration depends on a peculiarity of fluid 

 motions derived from steady motion by the addition of a 

 motion of translation, which 1 have not seen noticed. In 

 such motions the displacement, say f, perpendicular to the 



