Manchester Memoirs, Vol. l. (1906), No. 8. 3 



case tends. An attempt is made to shew (and I hope the 

 argument is sound) that all progressive waves in deep 

 water are of the class which possesses a horizontal series 

 of poles above the surface of the water, and that there is 

 no limit to the closeness to the water surface wh ich this 

 series of poles may assume. As this process is continued 

 the point of inflexion of the wave surface continually 

 approaches the crest, and in the final stage coincides with 

 it, and the wave profile shews a finite angle at the crest. 



In this critical case, the wave-profile is the same as 

 that investigated by Mitchell * in his paper on " The 

 Highest Wave in Water." 



The Solution of Lagrange's Equations. 



The functional solution of Lagrange's equations for 

 irrotational motion of a fluid in two dimensions used in 

 this paper is found as follows. 



Since the equations of condition may be written in 

 the form 



■^(x + iy) ~. ix + iy) 

 ^ M ^/ •^ 



= 0, 



x-\-iy must be a function of x—iy and /, and in the case 

 of steady motion which I am here considering, a function 

 of x—iy only. 

 Writing 



2^{x + iy) = (l>\x-iy), 



we get 



^'{x + ty)^fx + iy) = ^{x + iy)f'{x - iy) 



= a real quantity, 

 * F/ii7. Mag., November, 1893. 



