Highest Waves in Water. 437 



The form of the wave is shown in the diagram. The error 

 in (fly near the summit means an error of about 1*5 per cent. 



in x,y, and g, due, as before stated, to the slower convergency 

 of the series for dic/dz at that point. This does not affect 

 the determination of the height of the wave. 



Velocity of the Wave. 



The length of the wave is 3*96 and 2^=1*66, so that, in 

 general units, if L is the length of the wave and V its velocity, 



V 2 = -191#L. 

 For the infinitesimal wave, 



V 2 = <7L/2tt. 



The ratio of the velocities of the highest wave and the in- 

 finitesimal wave is therefore 1*2. The wave observed by 

 Abercromby (loc. cit.) of length 765 feet and height 46 feet 

 had a velocity of 47 miles per hour ; the highest wave of the 

 same length would be nearly 100 feet high and would have a 

 velocity of 47 miles an hour very nearly. 



Water of Finite Depth. 



Here the poles of the function dTJ/dw have to be indefinitely 

 reflected in -v|r = K', the bottom of the water, and i|r = 0, the 

 surface, in order to make U real or 6 constant at the bottom. 

 The singular points in the whole field are therefore at 



w=2nK + 2n f iK!, 



tt' = 2ftK being the wave summits. 

 We are thus led to the form 



-j-=H 3 (w)e*K < l + a l cos^-{w—iK l )+ ...>, 



where H is Jacobi's function so denoted. This equation is 

 susceptible of the same treatment as the simpler case. 



University, Melbourne, 

 August 7. 



