24 GWYTHER, Range of Progressive Waves in Deep Water. 



also proposed to find a connection with the process 

 employed by Mr. Mitchell. 



For this purpose, I differentiate with respect to //, the 

 equation of surface condition (3), and obtain 



^\^{iku)-if{-iku^^^e^\^{iku)<^{-iku)\^ . . (37). 



In the first case, I write 



and obtain 



fiiku) = [\P(iku)y\ 



miku)\i-[^-tku)\^ 



2 du 



(38). 



On rationalising the left-hand side of this equation, 

 the right-hand side will be found also to have become 

 rational. Thus 



■i^'Jku) -\p(- ikii) 



jiu 



., .d 



"§,' — 



du 



\^{iku)-^{- iku)\^ 

 i\){iku) \p( - iku) 



(39). 



It would appear that a continuation of the terms in 

 (29) could be most readily obtained from the condition 

 in this form. 



The stage most consonant with the final form of the 

 result will be reached by writing 



^{lh^) = {l- 4V^")-V^1 



(40). 



and <p{ - iku) = {i - (^e- '"'")-' jV\ 



where U and V are rapidly converging series of the types 



