﻿A 1 *- 2 



A 2 =^~ ra 



Dr. J. R. Wilton on Ripples. 693 



From these equations, remembering (3), I find by 

 successive approximation : — 



1 30k 3 -71k 2 +17k-S , 



, ... (iu) 



A 3 



A 4 = 



2 2*-l 16 (2a:-1) 3 (3/c-1) 



3 2/e 2 -ll* + 8 3 



16(2*-l)(3*-l) a ' 



+ 



13248^-53640^ + 63260^-29010^ + 7971/^-1216 



768(2*- 1) 3 (3a:-1) 2 (4/<;-1) 



18* 3 -183/e 2 + 361*;-128 



48(2k-1)(3k-1)(4k-1) 



a\ 



(12) 



A 5 288/^-4680^ + 18980*; 3 -24786tt 2 + 11091*;- 1600 



5 ~ 1536 



1 1 



°~ 2 2 /C+ l6 



(2/c-l) 2 (3/c-l)(4/t-l)(DA;-l) 

 1 2/c 2 -15a; + 16 , 



a 5 , 



1 24^+220a; 4 -2422a; 3 + 4701/ C 2 -2858/c+704 



256 



{2tc-iy(&tc — l) 



12* 2 + * + 8 2 

 o Ak—L 



1 24a: 5 -164/c 4 -566/c 3 + 1821^-1322^ + 448 4 

 + 128 (2/c-l) 3 (3«:-l) a * ' * ' 



I have also calculated, independently, the values of these 

 constants in the particular cases k = 1 and # = 2. 

 When /c=l, A x = —a, 



A 2 =-ia 2 + a 4 , A 3 =^a 3 +^a 5 , 



A -H^ A- 3535 a 5 



11 2 . 241 4 



8 25b 



(16) 



When k=2, A : =— a, 



* X i a 3 3 547 5 



A »-12T, a » A3 "40 a - 67200*' 



A 4 = 



840 "' 



A* 



47 



,.cr 



t> 1 Q •> 



2 8 



a 80 ' ^ =, ' _ J"" 



40 



(17) 



(11) 



(13) 



(11) 



(15) 



