due to a Travelling Disturbance. 545 



To calculate f'(6) we have 



f'(0) _ 1 d K ^sin 6— y cos 6 ,«^ 



*7(0) ~~Kd6~ xcosO + y sin ' * " * * ' 



and thence, having regard to (25), 



/"(g)- i-^R m> 



We may therefore write in (27) 



/"(0)=-2»7r(l + *J*->). . . . (35) 



5. As a first application of our results we may take for 

 simplicity an imaginary case, which has, however, already 

 been used for illustrative purposes *. A medium is assumed 

 to be such that the group-velocity in it is constant, so that 



(r=(T +m (36) 



Hence 



k= ^— rr , (37) 



c cos - U ' 



showing that the wave-system is limited by the condition 

 cos#>U/c. The equation (32) which determines the con- 

 figuration of the wave-ridges becomes 



;>=^( C COS0-U) (38) 



The ridges therefore form a system of circular arcs, of 

 radii 27rnU/o- , lying to the right of and convex to it, 

 and touching two straight lines drawn from at angles 

 + sin _1 (U/c) to the axis of x. 



6. In the case of a pressure-point moving over the surface 

 of deep water we have, if capillarity be neglected, 



«=;=7> (39) 



COS 2 



The equation of the " isophasal " lines, as they are called 

 by Kelvin, therefore takes the form 



p= cos 2 6. ..... (40) 



* Schuster, < Optics,' 2nd ed. pp. 333-4. 

 Phil. Mag. S. 6. Vol. 31. No. 186. June 1916. 2 



