4 GwYTHER, Range of Progressive Waves in Deep Water. 



and therefore 



^(.v + iy) = u + ib, 



where u is real and b is independent of t. 



If we suppose now that this relation can be reversed, 



we may write 



x-\-iy=f{u^ib) . .' . (i). 



This is, of course, exactly comparable with the 

 Eulerian relation usually written 



x^iy=f{<p + i-^). 



Proceeding again with this relation, we have 



d . (ill 



-{x + ty)=^f{n-,tb)-. 



This must be a function of x—iy, and therefore of u — ib, 

 and it can only take the form cj f\ii — ib), where c is an 

 absolute constant. 



We therefore must have 



f\u + ib)f' {u - ib)du = cdt 

 and 



Jf{u + ib)f'{u-ib)dii = ct + a . . (2). 



This relation gives, in any case, ti as a function of a and 

 b, and these symbols in the Lagrangian equations are now 

 introduced as constants of integration. 



I now proceed to show that continuing to use the 

 Lagrangian method, the expression for the pressure takes 

 the same form as that derived from the Eulerian method. 

 This is solely a matter of form, since the result could not 

 be otherwise. 



The equations with which we have to deal are 



bail, ^^ -^^ -^ ' ) dci. ca -^ ha) 



I' 

 We have 



etc. 



X- +y^ = cu , 



• ex -ly lu 

 X^r + Vtt- =Ct~ 



ca ca da 



