Chta-Shun Yth 
Eliminating er between (88) and (90) , we have 
! 
a OP ee ee 
000 Oir\. (91) 
! ! 
(P9¥'o9 0F? 000 
or io 
= (| ! ' ea = ! = 
=o op) £ (opts ) AB 'o |¥o0 Pl 
The boundary condition” at y =0 is 
If the upper surface is fixed, the boundary condition there is 
Vo 96d) = GE (93) 
If there is no flat upper surface, and the conduit is full of fluid, (93) 
can simply be applied at the highest point. On the other hand, if the 
upper surface is free, the boundary condition (82) can be written 
as 
a Ee [2v9(@ . Vo9( | ; (94) 
Integrating (90) in the Stieltjes sense over a vanishingly thin layer 
at the free surface, we obtain 
_ £'(d) 
2 B%20(9 = Fa) Yoo(%) 
Substituting this into (24), we obtain 
f fi(d) 
are i - | Yoo) » (95) 
*If there is a flat rigid upper surface the boundary condition 
there for v is v5,(d) = 0, which can be satisfied only if 
20 20 
f'(d)=0. The present analysis for symmetric channels is there- 
fore valid only if the channel boundary has no flat rigid part. 
1420 
