a Single Impulse in Water of any Depth. 253 



so that we have 



m=t{f(ix) +/*/»} =Vt .... (3), 

 where 



V=/W +/*/»* W; 



and we have by Taylor's theorem for m— p very small, 

 m[ X -tf{m) ] =A*[*-i/M ]-*[>/'» + 2/V)]i (m-/*) 2 (5) ; 

 or, modifying by (3), 



»[*-#»] -woo + [-/*/"<» - 2/vmo-/*) 2 } (Q. 



Put now 



m ^[-/*/%*)-2/W ' * " ( j ' 



and using the result in (1), we find 



« = 0[-/*/"(^)-2/V)] J " ' ' K> ' 



the limits of the integral being here -co to co , because the 

 denominator of (7) is so infinitely great that, though ±a, the 

 arbitrary limits of m— fz, are infinitely small, a multiplied by 

 it is infinitely great. 

 Now we have 



j da cos a 2 = l da sin a 2 = a / - . . . (9). 

 Hence (8) becomes 



cos[^y>)]- S in[^y(/x)] _ y2 C os[>y( /t )+frr] 

 *[-«TM -2/»? - *[-/*"&») -2/»]* ^ 



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

 point of the group, remark that, by (3), 



^V0i)-#»[*-.*/»], 



and therefore the numerator of (10) is equal to v/2cos#, 

 where 



0=H>-*/(/*)]+iT .... (100, 



and by (2) and (3), 



d/d^lx-tf ((*.)]}=<); 

 by which we see that 



de/d,v = fM, and d0/dt=-fif(fi) . . (10"), 

 which proves the proposition. 



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

 tion of Prof. Stokes's original result. 



