Change of Form of Long Waves. 429 



Sg = — j- Bex, in this velocity, but, on taking the variation of 

 (12) with respect to a, we obtain 



dt I ~ftx ~ y ' "ftx 



which equation may be easily verified geometrically. 



II. Stationary Waves. 



OjTI 



For stationary waves y-must be zero. Therefore we have 

 from (12) dt 



This gives by integration 



o,+W+i^+i^=0; (14) 



and by multiplication with 6 drj and further integration, 



c 2 + 6c lV ±y* + 2*y 2 + *(^J = 0. . . . (15) 



If now the fluid be undisturbed at infinity, and if I be taken 

 equal to the depth which it has there, then equations (14) 



and (15) must be satisfied by ?7 = 0, —-=0, and ^-— ^ =0. 



Therefore, in this case Ci and c 2 are equal to zero, and equa- 

 tion (15) leads to 



g_ ±v /r?s±H .... „«, 



Here, before we can proceed, we have to discriminate between 

 a positive and a negative. In the first case 2a is necessarily 



negative because =r— must be real for small values of n. If, 



then, we put it equal to —A, we have 



from which, supposing oc to be zero for rj = h, we easily obtain 

 the well-known equation of the positive solitary wave, viz. : — 



?7 = Asech 2 .^A/_. ...... (17) 



