20 GWVTHER, Range of Progressive Waves in Deep Water. 



From these we obtain, writing Jie~^'' for H, as the 

 equations corresponding to (24), ('25), and (26) : — 



ku = ^i-\ ihe-"" + zh'e-'" + ir'iy-'""' + ^^^-''*'')1 sin ^ 



- \h\e-"^" - il-^-*') + ii'i^^-e-""' + if e--^^" - ^tf-«^'')|sin 2/i 



+ sV^-tHs^"''' - 6^-"') sin 3;z - ^l^h\2oe-'"' - i y^""*") sin 4/i 

 At = ;x - J/^^-^* + |;r^-^^'' + h\^e- ■«•* + >/^^-^^*)! sin ^t 



- {hWe--"' - le-'"} + y^«(f 1^--^* + ^e-'"' - W-^-"^*)] sin 2^ 



- ^\/i%6e-''"' - e-'"') sin 3^ - -^h^/i%ioe-''' - e'"''') sin 4^ 

 ^_y = Z'^ - \/ie-"' + ^/fe -"■' + /I'ip-''" + -V-^-«*")l cos fx 



- {/I'iU-"-'" - y'''") + /iM'^e--'" + U-'" - -^''e-''')} cos 2^ 



- ^V^^(2^-^*" - e-''") cos 3,u - ^l^/iX^e-''" - 2e-""') cos 4ju. 



(28). 



From these we may evaluate 2gy — ai, to determine 

 the law of variation of the pressure downwards, and so 

 measure the error in Gerstner's assumption that the 

 pressure is uniform along each stream line. The 

 expression is, however, very cumbrous, as might be 

 expected. 



The Approximate Determination of the Extreme 

 Value Permissible for h. 



When the values of //,, Z/^,... from (7) are substituted 

 in f\u + ib), the successive terms in the expansion have 

 the general appearance of those of a divergent series. If, 

 however, attention is confined to the leading terms in the 

 coefficients in the series, it may be noticed that there is 

 a marked resemblance, in point of divergence, with those 

 of the Binomial expansion of (1-3//^"")"^ In order to 

 bring into prominence and to test the extent of this 



