26 GWYTHER, Ranoe of Progressive Waves in Deep IVater. 



This becomes, in the special case when 4 = 1, identical, 

 except for the difference in units, with the equation 

 which Mr. Mitchell has employed to obtain the circum- 

 stances of the Highest Wave in Water. 



We may, however, proceed to rationalise the ex- 

 pressions as in the previous case, and thus obtain 



This may be used when 4 is less than unity to determine 

 the coefficients to the expansions of [/ and V, and also, 

 when 4=1, after division by (i - <r^''*")*(i - 4^-"")*, it may 

 be conveniently made use of for the same purpose, since 

 the series on both sides would proceed in cosines of 

 multiples of the angle ku/2. 



By the theory now proposed, the form of the solution 

 IS the same whether for rounded or for the critical pointed 

 wave. In both cases the solution shows poles, which in 

 the case of critical pointed waves lie on the surface of the 

 fluid, but in the case of the rounded waves are situated 

 above the surface at a less or greater distance according 

 to the height of the waves. 



It has not been shown that the equation of surface 

 condition has a solution of this special form, but merely 

 that the terms of the solution in series found by another 

 method submit to a great simplification when treated in a 

 special way. 



This being the case, a conclusion in regard to waves 

 under other circumstances would not be justified, but it 



