Manchester Memoirs, Vol. l. (1906J, No. 8. 5 



• Ix • cy ?u , 



-^T^ +J'?7 = ^^7 5 SO that 

 CO -^ to do 



c {p 1 • "> , ^ ^ '"i/A ^ 



ca(p *' - J ^//V W 

 with a precisely similar equation. 



The condition that the motion shall be irrotational 

 requires that c shall be independent of a and b, which is 

 here satisfied since c is an absolute constant. 

 The pressure is given by 



P 1 • 



-xy + hcu = constant 



P 



or 



~-gv+—fr, — . .,. .,, r^^ = constant . . (3). 



p ^- 2f{u + i0)f{u-w) ' ' 



It follows from this that if x + (}'=/{(p -{- t\p) is a 

 solution of the Eulerian equations which can be made to 

 satisfy the conditions of a steady motion, then 



X + iy =/ {u + ib) 

 where 



J f {u + ib)f\u - ib)dii = c/ + a , 



is the solution of the Lagrangiari equations under the 

 same circumstances. 



Application to Stokes' Problem. 



The mode of treatment of waves of finite amplitude 

 in deep water employed by Sir George Stokes in his 

 " Supplement to a paper on the Theory of Oscillatory 

 Waves " * leaves the question in the form to which the 

 method of the last chapter is applicable. 



Confining myself to the steady motion, from which 

 the progressive wave is to be obtained in the usual way, I 

 write, following Stokes, 



X + iy =/(7i + ib) = u + ib - /v^^'»*("+'*) . . (4), 



* Mathematical and Physical Papers, vol. i., p. 314. 



