442 Drs. Korteweg and de Yries on the 



degree of accuracy. Now such a series may be obtained in 

 the following manner. Let 



tt=Aicn f i* •«(*!+ iy (M=^/ r ^ F ) • 



(60) 



represent the solution of an equation 



Vi 2 = ar h (h 1 —y 1 ){k 1 + r] l ), .... (61) 



where \ and k x have values which are slightly different from 

 those of h and k in (39); then these values and the coefficients 

 a, ft ', &c, of a series 



v = a Vl + Pr) 1 2 + c rni * + $r) 1 4 + (62) 



may be determined in such a way* that this series (62) 

 satisfies the equation (39). 



Indeed, substituting (62) in (39) and taking into account 

 (61), equation (39) reduces to 



(« + 2^ 1 + 3 7 t ?1 2 + . . nh-V l )(h+Vi) 



= {* + ftvi + Y*7i 2 + • • •)(/* — ur) l — ftirf — 7^i 3 + ...)(& + otrji 



+fai*+yni + ...){l+b* Vl + (bft + c*)yi l + ...), 



and it is only necessary to equalize the coefficients of the 

 corresponding terms of both members of this equation. 



If we retain all terms to the fourth order, we find in this 

 way, after some reductions : — 



ah 1 k 1 -hk=0- (63) 



^(h 1 ^k 1 )-^(h-k)-{b^-3^hk=0. f ..... (64) 



-a 3 + * 4 -{ba*-2* 2 ft)(h-k)-(cx 3 -2ba?ft+8ft*-Qay)hk = (65) 



— 4a/3 + 3a 2 /3 + bct-{c<x* + ?>ba 2 ft- 3/3 2 -4<*y) (h -fc)=0 . . (66) 



-4;ft 2 -6ay + c** + Ma?ft + 3<xft2 + 3u 2 y=0. • . . ... . . (67) 



To a first approximation these equations are satisfied by 

 taking 



h = h', k 1 =k; « = 1; ft = b; y = b* + ±c . .(68) 



If then we substitute in (63), (64), (65), and (66) 



/^rrA-j-e, k 1 = k-bce, u = l + a u ft = b + fti 



where en 1 and ft A are quantities of the first, e and ce of the 

 second order, we find from these equations by second approxi- 

 mation: — 



* The coefficient a in (61) might also have been chosen slightly 

 different in value from a in (39), but this would only have introduced 

 an unnecessary indeterminateness in the solution, 



