[ 308 1 



XXIX. An Appendix to the Paper on the Classes of 

 Progressive Long Waves. By R. F. G-wyther, M.A.* 



ri^O make a rigid test of the method of approximation which 

 JL 1 have proposed in my paper on " The Classes of Pro- 

 gressive Long Waves " f, it appears to be desirable to pursue 

 the investigation to a higher order of approximation, espe- 

 cially in the case of the Solitary Wave. For this purpose I 

 continue the expansions in the paper referred to, in which 



(c 2 - 9 ~) h 2 f" = - 3c/' 2 + 2(c 2 -gh)f 



gave the first approximation. 



From these terms we gather the mode in which the order 

 of the terms is to be estimated; namely, that /' is of the 

 order {c 2 —gh)/c 2 , and/'" of the order (c 2 — gh) 2 /c 4 . I shall call, 

 in order to avoid fractional orders,/' of the order 2 of small 

 quantities,/" of the order 3 J, and so on, 



Acting on this principle, the terms of (3) of the previous 

 paper (the equations in this paper are numbered continuously 

 with those of the former) can readily be arranged in sets of 

 orders 4, 6, and 8. We thus obtain 



( C 2_^W, 



[- 2 (7 c 2 -gh) 2c 2 -gh cV c 2 + 2gh „„ 



+ L 7! n/ 12c hU T 7/ 12c ' 



+^/r 2 -^- 3 /T"]&c. 



= -3 c /' 2 + 2( C 2 -^)/' 



+/' 3 (13) 



The character of the integrals of this equation to the 4th, 

 6th, and 8th orders are then seen to be as follows: — We shall 



be able to express in each case 2 (c 2 — ^)h 2 f" 2 in ascending 



powers of /'. The terms of the expansion will proceed as far 

 as the third power in the solution to the 4th order; as far as 

 the fourth power in the solution to the 6th order ; and will 

 extend to the fifth power in the solution to the eighth order; 



* Communicated by the Author. 



t Phil. Mag. August 1900. 



% This was overlooked in forming the faulty solution (9). 



