a Free Procession of Waves in Deep Water. 115 



than several times the period and not less than several times 

 the time one of the wave-crests takes to travel through a 

 space equal to I, be independent of the particular forms of $t 

 and F. From the theory of Stokes, Osborne Rejmolds, and 

 Rayleigh, we know that it advances at half the speed of a 

 wave-crest ; but their theory, so far as hitherto developed, 

 does not teach us the law according to which the front, as 

 it advances, becomes longer and longer in proportion to s/t, 

 nor even the fact that it does become longer and longer. All 

 the details of this interesting question are explicitly given 

 in what follows : having been found with great ease for the 

 particular case, 



F<» = 0, and gO) = [ v ^ + ; 62 | . . (11), 



where b denotes a length of any magnitude, which we shall 

 take to be very small in comparison with 27n//o) 2 , the wave- 

 length. We shall in fact find that 



P^^l^^Y^. . . (12), 



in the particular processional case of the general equations 

 (1) . . . (6), which we now go on to work out. 



Remembering Cauchy and Poisson's discovery that every 

 surface of particles which are in a horizontal plane when un- 

 disturbed, fulfils the condition of a free upper-surface (so that 

 if all the water above it were annulled the motion of the water 

 remaining below it would be undisturbed,) in the case of free 

 waves of infinitely deep water; we see that when _p f 0) = const., 

 we have also, in our notation, p = const., for every constant 

 value of y. Hence, looking to (3) above, we must find, in 

 the case of free w r aves, 



~di~W (ld) ' 



for every value of y, and not only at the upper surface, y = 0. 



Thanking Cauchy and Poisson for this as a suggestion, but 



not assuming it without the proof of it which we immediately 



find ; and borrowing now from Fourier * his celebrated 



" instantaneous plane-source " f solution of his equation 



dv d v 



-j- = k -r-£ for thermal conduction, assume, as an imaginary 



* Theorie Analytique de la Chaleur. 



t W. Thomson's Collected Papers, vol, ii. p. 46. 



