Deep Water Waves. 389 



The leading term in each coefficient A n may be written 



down. In fact, 



n n ~ 2 a n 

 A n = ( — ) n _i T~i + higher powers o£ a, 



where A x = —a. 



For, if we retain only the first approximation to each 

 coefficient, we shall have, in order to determine A» from the 

 values of A 1? A 2 . . . A H _i, supposed known, the equation 



(M-l)A» = n(A l A n _ 1 -hA 2 A ;i _ 2 + A 3 A n _3+ . . .), 



where the last term on the right-hand side is JAj* if n is 



even, and A'n-i A K+ i if n is odd. We have also put C= — J, 



•J 2 



its approximate value. 



The solution of this system of equations is easily seen to 

 be, assuming A x = —a, 



n n ~ 2 a n 



The most direct proof is obtained by expanding x and x 2 in 

 powers of xe x by Burmann's theorem, squaring the first 

 result, and comparing with the second. 



I have not been able to find a general formula for even 

 the second order approximation to the value of A n . It is, 

 however, easy to derive a sequence formula by means of 

 which the second order approximation to any given co- 

 efficient may be calculated. In fact, if we retain terms of 

 order r+2 in the equation for A r , we obtain 



0[2rA r + 2(r+l)A 1 A r+1 ] +A r + HMr-i + A 2 A r _ 3 



+ . . . A r _iA 1 ) + A 1 [rA 1 A P +(r-l)A 8 A r _ l + . . . A r A 1 ]=0, 

 i. e. 



(r-l)A r -ir(A l A r _ l + A 2 A r _ 2 + . . . At-iAjl) 



+ 2ra 2 A r -(r + 1) A x A r+1 = 0. 

 If we put 



we obtain the following sequence equation for B,. : — 

 (r-l)B,-rfB r _ 1 + B r _,+ 5B r _,+ ?B r _ 4 + . . . + fcrSj^B,] 



L A 6 (r — Oj ! J 



= !,[(,+ iy_,v]. 



In the same way, if we put 



(->" A "= /,'._"','',. +KX + -' + Ox +1 , 



