2 



, as 



438 Drs. Korteweg and de Vries on the 



Figs. 5 and 6 refer to the equation 



. . "2lTX , . . 477U' 



77=^ sin — V ^AiSin -^— • 



A, A 



A / I \ 



In fig. 5 -p is supposed to be small compared with i — V 



is the case with waves of extremely small height. In fig. 6 



/ 1 \ 2 A 



we suppose 1 — 1 to be small in regard to -y. Generally for 



more complicated forms of wave these two cases have to be 

 discriminated. When there is a moderate proportionality 

 between the two fractions the result is still more complicated. 

 Finally, fig. 7 refers to the equation 



V = Ai sin — 4-Aj sm -— > 



A A 



I 



in case 



ase that (— ) is the smaller fraction. 



It is worthy of remark that all these waves grow steeper 

 in front. 



V. Calculation of the Fluid Motion for Stationary Waves to 

 the Higher Order of Approximation. 



In order to remove every doubt as to the existence of 

 absolutely stationary waves, we will show how by develop- 

 ment in rapidly convergent series the state of motion of the 

 fluid belonging to such a wave-motion may be calculated. 



Expressing again the horizontal and vertical velocity of a 

 particle by means of the series (1) and (2) which fulfil all 

 the conditions for the interior of the fluid, we have only, 

 neglecting capillarity, to satisfy the surface-conditions, 



*i = ^, ....... (37) 



and u* + Vi + 2grj = constant (38) 



For the case of cnoidal waves, which is the general one, 

 we have found as a first approximation, 



But now, to obtain higher approximations, we assume, indi- 

 cating by accents differentiation with respect to #, 



i/*=a^A— i|)(*+ilX 1 + 6 * + ^ + -- •), • • (39) 

 and 



/— g + rr] -f srf 4- tvf + urf -f (40) 



