1887.] Waves produced by a Single Impulse. 81 



so that we have x = WO*)} = yt, (3) 



where </ =/00 +W0*) 5* (4) 



and we have by Taylor's theorem for m—ja very small : 



m[ a5 -^(m)] = ^-^( / x)]-M^/''W + 2/X^)](^-/4)2, . (5) 



or, modifying by (3) 



»[•-*/(»)] = «{^/W.+i[- i »/"W-2/W](™-^) s }. ■ (6) 



Put now m _, = __^__ (7) 



and using the result in (1), we find 



4t ~ <Hf-/«/'0»)-2/W]* ' ( 8 ) 



the limits of the integral being here — oo to oo, because the denomi- 

 nator of (7) is so infinitely great that, though +«, the arbitrary 

 limits of m—fi, are infinitely small, x multiplied by it is infinitely 

 great. 



Now we have f^da cos a* = f°°^dcr sin (r° = -v/Ci"") • • • • (9) 

 Hence (8) becomes 



_ COS [^/Qi)]-Sill[^8/W] _ y/ 2cQS[>y(*0+frr1 n fn 



To prove the law of wave-length and wave-velocity for any point 

 of the group, remark that, by (3) 



and therefore the numerator of (10) is equal to a/2 cos 0, where 



= M loc-tf(fi)-]+i r , (10') 



and by (2) and (3) ~{^[x-tfM]} =0; 



by which we see that 



d9jdx = fi, and dOjdt = —/if (ft), .... CIO") 

 which proves the proposition. 



* This is the group-velocity according to Lord Rayleigh's generalisation of 

 Prof. Stokes's original result. 



G 2 



