Manchester Memoirs, Vol. xliv. (1900), No. <». 3 



Fas ordinate, the i^roup-velocity will be represented by 

 the intercept made by the tangent on the axis of V. Thus 

 in Fig. I, FN represents the wave-velocity for the wave- 

 length ON, and OF represents the group-velocity. The 

 frequency of vibration, it may be noted, is represented by 

 the tangent of the angle PON. 



Some particular cases may be noticed. 



(i) In the case of gravity-waves on deep water, V^^^- ; 

 the curve has the form of the parabola y- = 4a,v, and 



0F= -FN, i.e., the group-velocity is one-half the wave- 

 velocity. 



(2) For capillary waves, without gravity, FocA"-, and 



■2 

 the curve has the form xy- = a^. In this case 07^= -FN. 



(3) When both gravity and capillarity are operative, 



where a is a certain constant, and the curve has the form 



1'- .V a 

 b'^ a X 



This is shewn in Fig. 2. The curve indicates at once the 

 existence of a minimum wave-velocity corresponding to 

 X = (? ; also that for any prescribed wave-velocity greater 

 than the minimum there are /zao possible wave-lengths, of 

 which one rapidly increases, whilst the other diminishes, 

 as the wave-velocity increases. It appears, moreover, that 

 the group-velocity is less than, equal to, or greater than 

 the wave-velocity according as the wave-length is greater 

 than, equal to, or less than the critical wave-length a. The 

 frequency on the other hand, steadily increases as the 

 wave-length diminishes. All these are of course known 

 results. It may also be noted that since two tangents can 

 be drawn to the curve from any point on the axis O V 



