440 I)rs. Korteweg and de Vries on the 



If now we write, in accordance with (37), 



Wl =-}=A + Bi7 + (V + W+..., . . . (45) 

 V 



we have from (43) and (44) : — 



A = q- larWik + &a*rPhk (h - k) = — rl + £aW 3 {/i - A;) 



+ labrlVik + iasi: s hk—^Qa 2 rl 5 (h — ky + 8 3 Qa 2 d d hL . (46) 



B = r — ^arl\h — k) — ^abrlVik — %asl 2 hk — \arllik 

 + Jja W (A - &) 2 - fofrPhk =-2sl-r- ±arl* 



+ \abrl\h-k) +%asl^h-k)+iarl%h-k) + ±a?rl 5 (h-k). (47) 

 C = s + fa/ 2 >'-ia6W 2 (/<-#)-2a^ 



= -3^-25-aW 3 -fa^ 3 -|arZ 2 —3%aVZ 5 . . (48) 

 D:=t+afrP + $a«P+4aW + Aa*rP (49) 



Moreover, since (38) may be written in the form 

 ihXl+v' 2 ) + 2^ = (A + Bt7 + C^ + D7? 3 ) 2 (l + a/^ 



+ a(A— A:) rf l —arj^) + 2gy = constant, (50) 



we readily obtain 



2AB + aA&A 2 + 2# = 0, .... (51) 



2AC + B 2 + a(/i-£)A 2 = (52) 



2AD + 2BC-aA 2 =0 (53) 



_ From the equations (46), (47), (48), (51), (52), (53), the 



six quantities q, r, s, t, a, and b may be calculated,, and if we 



had retained everywhere terms of one higher order, we might 



have got eight equations with eight unknown quantities, &c. 



By a first approximation we readily obtain from (46)~(49) : — 



r= — 2; s=^+iaql; t= — ^ d —±aq + ±abql — £- 8 a?ql d ; 

 A = q; B=— 2; C=^— iaql ; 



D = - y + %aq-%abql + fypql* ; 



and then from (51)- (53), 



3 3 

 q*=gl; a= p ; b= -^ (54) 



