Mr. J. M e Cowan on the Solitary Wave. 55 



The path of any particle is given by (39), (41), and (43), 

 but if we neglect t, we can at once eliminate x' from (39) an 

 (41), and obtain 



(Z-a) 2 + Z 2 + 2aZcotm(z + £)=a'>. . . (45) 



If we expand this, neglecting £ 2 as it is of the same order as 

 a 3 , we get 



(g-a) 2 + 2a£cotmz = a 2 .... (46) 



as an approximate equation to the path described by any 

 particle originally in the plane z. To this order, therefore, 

 each particle initially at a distance z from the bottom describes 

 that part of the parabola given by (46) which lies above the 

 level z. This gives 2a for the maximum horizontal displace- 

 ment, and -J a tan mz for the maximum elevation, of a particle, 

 but more exact values have already been given in (44) 

 and (40). 



9. The Energy of the Wave. 

 The potential energy of the wave per unit breadth is 



J 00 /-»00 



ifdx=gp Tj^dx; 

 -oo Jo 



but, by Lagrange's Theorem, (11) gives 



. „ a 2 sin 2 mh , 2a 3 d sin 3 mh 



1 2 (cos mh + cosh mxf (3 dh (cos mh + cosh mx) 3 



Hence by (29) and (30), 



y=rigpma 2 h 2 (l+^ Tj ma) (47) 



The kinetic energy per unit breadth is 



the integration extending throughout the liquid. 

 Thus 



T = ±pfi{ (U + u) 2 + io 2 \dx dz, 



= ipjJlPtf/e dz + pJJUw dx dz + ipfofdx dz, 

 = \pW§dx dz + P XJ$df dx + y§d<f> df. 

 .-. by (4) and (5), 



T = ipU%v-2ah); (48) 



or, using the approximation (31), 



T=lpU 2 ma 2 A(l+£ma). 



