96 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
the process and find the fluid motion which has a given free surface, that 
is to say, a given surface along which the pressure and velocity are constant. 
We proceed as follows ; — 
z = x + iy ....... (2) 
m 
= j ds(cos 0 + i sin 0) 
= jdsd^ . . , . . . . (4) 
c 
If the constant in (1) be made Po~\-p/^, where is the constant pressure 
along the free stream line, we may write the velocity 
= 1 
ds 
(5) 
.'. (p = s ii (p = 0 when s = 0. 
Hence, if ^ = 0 be the free stream line whose intrinsic equation is 
F(6‘, d) = 0, we have merely to substitute for s. 
It follows, therefore, that 
z = jdtvd® (6) 
®) = 0 (7) 
where w = (j) -\- iyfr , and @ = complex parameter, represents a fluid motion 
with E(s, 0) = 0 as the equation to the free surface. Any one of the other 
stream lines may then be taken as one of the rigid boundaries. Some 
very interesting problems of the motion of a fluid against a fixed obstacle 
