and the initial condition <j), = 1 = at t = 0, instead of the conditions i = on 

 S at t = and t = T with large T used in the previous section, we obtain, for any 



time, x 



6 J *1 * *ltt dx = J 6 * 1 ( x .t) (f) ltt (x,T-t)dxdt + * 1 (x,t) 6(J> ltt (x,T-t)dxdt 

 Fo Fo Fo 



= 2 6ct) 1 (x,t) 4> ltt (x,T-t)dxdt 



s w 

 Fo 



+ {((^(x.t) 6<fr lt (x- t t-T)-$ 1( .(x,.t) 6(J> 1 (x,t-T)}^ =() 



S Fo 



J J e^Cx.t) <j) ltt (x,T-t)dxdt 



S v 

 Fo 



6(|> 1 * *ltt dX (16) 



2 



Fo 



where T need not be large, and x represents a point on the free surface z = 0. Thus, 

 we obtain as in Equation (12a) 



6J = - J J V 2 * x * 6(J) 1 dzdx -I S$ 1 * ($ lr + | cj) ltt ) dx 



IFo 



ri AJkl - h 



1 V In x t 



(<j> -<J> ) * 64>„ dz - (+. -H|» ) * 6<p d z (17) 



J 1 z in J In 2n 1 



S J S J 



