﻿G92 Dr. J. R. Wilton on Ripples. 



To obtain closer approximations we must expand the right 

 hand side of (2) and re-arrange in cosines of multiples of f, 

 The coefficient o£ cos n% is then to be equated to zero. It 

 would be possible to obtain the general form of the equation 

 thus derived, but it would be extremely complicated and 

 there would be no advantage in doing this. We shall there- 

 fore write down the resulting equations only in so far as 

 they are necessary to obtain the approximation we desire. 

 We shall determine each approximation accurately so far as 

 T is concerned. 



The equations derived from (2) are, if we retain terms of 

 the fifth order and reject those of higher orders, 



^ + 2C(l-fA 1 2 + 4A 2 2 ) + 2A 1 2 + 4A 2 2 + 4A 1 2 A 2 

 + <A 1 2 +gA 1 4 -2A 1 2 A 2 + 8A 2 2 ) = 0, ... (4) 

 2C(Ai + 2AiA 2 + 6A 2 A 3 ) +A X + A x 3 + ZA^ + 5A 2 A 3 + 6A 1 A 2 2 + 3Ai 2 A 3 



+ <A 1 -|a i 3 + 3A 1 A 2 ~ gjA^-H I AM,- yA^-^iV+iSAaA,) 



= 0, . (5) 



2C(2A 2 + SAxAa) + A 2 + A x 2 + 3Ax 2 A 2 4- 4A X A 3 



+ / C (4A 2 -iA 1 2 +jA 1 4 -2A 1 2 A 2 + 6A 1 A 3 ) = 0, . . (6) 

 2C(3A 3 + 4A a A 4 ) + A 3 + 3AjA 2 + 5A X A 4 + 4A X 2 A 3 + 2AiA 2 2 



- A 3_Q A A —.— A 5 i 1& A 3A 1^ 



e-^-i — oj±iJ±2 190^1 -t -^- J\.i hl 2 -r 



-|a 1 A 2 z + 10A 1 A 4 )=0, . (7) 



SCA^ + A^AjAs + SA 



+«(16A 4 -^A 1 i -4A 3 2 + 3A 1 2 A 3 -6A 1 A 3 )=0, . (8) 

 10CA 5 + A 6 + 5 A! A 4 + 5 A 2 A 3 

 + «(25A 5 + ^A x »- ^Aj'A, + ^ VA, + ^AtV-lpA^-lSA^, 



=0. . (9) 



