430 Drs. Korteweg and de Vries on the 



In the second case 2a must be positive. In putting it equal 

 to h, and in substituting —9/ for rj, we have from (16) 



or, by integration, 



7}=—r)'=—h sech 2 x\ / . — 



V — 4<r* 



This is the equation of a negative solitary wave, and we are 

 able now to draw the conclusion that whenever <r is negative; 



/3T 



that is whenever the depth of the liquid is less than \ / — , 



V 9P 

 the stationary wave is a negative one. For water at 20° 0. 



this limiting depth is equal to 0*47 cm. (T=72, # = 981, 

 /3=0998 B.A.U.). 



Now, for a further discussion of equation (15), we drop the 

 assumption that the fluid is undisturbed at infinity. If then 

 / be taken equal to the smallest depth of the liquid, we must 



have ~^ = for w = 0, and therefore in virtue of (15) c 2 = 0. 



On supposing then a positive *, c x must be negative in order 



nn 

 that ^— may be real for small positive values of y, but then the 



equation 



?? 2 + 2^ + 6^ = (18) 



has a positive root li and a negative — k, and we may get 

 from (15) 



; ^ = ±\/lv(h-v)(k + v). . . ■ (19) 



By substitution in this equation (19) of rj = h cos 2 % and by 

 integration, we find 



'-^'V^VsSij? • • 



(20) 



* When cr negative, let then I be equal to the greatest depth. On 

 substituting <r= — o-', r)= —r{ we have again c x negative, 



and, finally, 



where h and — 7c are' the roots of ?/ 2 — 2ccrj +6^=0. 



