﻿698 Dr. J. R. Wilton on Ripples. 



same order. But if A 2 is o£ the first order, the second 

 approximation to (5) is 



(20+1)^ + ^^ = 0, 



a. e., C=-|-|A 2 ; 



and substituting in (6) we have, to the second order, 



-3A 2 2 + |A^ = 0, 



l. 6., ix 2 == + "2-txj. 



From equations (7), (8), and (9) we see that, if A x and A 2 

 are both of the first order, A 3 and A 4 are both of the second 

 order, A 5 and A 6 are both of the third order, and so on. 



If /e=l/3, it is easy to see that we may take the orders of 

 the successive coefficients to be 



1, 2 ; 1, 2, 3 ; 2, 3, 4 ; 3, 4, 5 ; ... . 



While in the general case, when tc=l/n, the orders are 

 1, 2, 3, ... n — 1 ; n — 2, n — 1, n . . . 2n — 3 ; 

 272-4, 2n-2, . . . 3n-5 ; 3n-6, . . . 



It is only in the particular case of ^/c = , i. e. n = 2, that 

 there is any ambiguity in the form of A„. 



Let us return now to the consideration of the case K — h. 

 To the first order we have 



Ax=— a, A 2 = ±ia. 



And, on substituting these values in the other equations, 

 we find as a first approximation, 



Aj=— a, A 2 =+^a, A 3 =±fa 2 , A 4 =0, A 5 = 0> 



C=-f + §a, /*=f±fa. 



As a second approximation I find, after rather long 

 analysis, 



A 1= =-a, A 2 =+Jra-|a 2 , A 3 =+ja 2 + ^ 5 a 3 , 



A 4 =+ia 2 , A 5 = 0, A 6 = + § §a 3 , 



C=-f + §a + - 3 V 2 , /^ = |±!«-!Sa 2 . 



In particular, when a = '2 the two sets of values are : — 



(1) A 1= --2, A 2 = -095, A 3 = -045, A 4 =--004, 



A 5 = 0, A 6 =--0003, 



fjb = l'5 ( J, c = 24*6 cm. /sec. 



(2) A 1 = --2, A S =-\L05, A 3 =-*015, A 4 = '004, 



A 5 = 0, A 6 = -0003, 

 /x=l-29, c=22-2 cm./sec. 



