22 GwYTllEK, Range of Progressive Waves in Deep Water. 

 These give approximately • 



a= 1-054, 



/3= - -068, 



7= - •016 . . . (31), 



and these lead to 



^^=■3514 -'023^"- -0054' . . (32). 



This value increases with increasing fractional values 

 of 4, and reaches the value '323 when 4 reaches the value 

 unity. 



Since the hypothesis of convergency would certainly 

 fail when Z, exceeds unity, we may take this to give 

 approximately the upper limit of the range of values 

 permissible for //. 



With these numbers we also find from (/j 



— = i + -i23r+-0374^+-oi9;^ , . {t,^)- 



and 



ka = '35i<C+ •0414^ + -oigii;'^ 



where 2a is the height of the wave from trough to crest, 

 and 



2a 



Y= -1124 + -oi 34"'+ -0064', . (34). 



if X is the distance from crest to crest of the wave. 

 This gives for the greatest value of this ratio 



20 

 ^-=•131 



approximately, a value not very greatly in defect of that, 

 viz., "142, found by Mr. Mitchell for the pointed wave. 

 With the same numerical values I find 



= (i - 'Ce'^'fli - (-018^- -0234'- -0054^'*" 



- ('oi 84" - •0094"')^''*" - (-0114^ - •0034^)^^'^" 



- -0064 V''^" - •0034^6'^'*" .... (35) 



