398 
MR. J. H. MTCHELL ON THE THEORY OF FREE STREAM LINES. 
On the Theory of Non-reentrant Free Stream Lines. 
The presence of sharp salient edges in a moving liquid always implies surfaces of 
discontinuity, but these may be closed or unclosed according to circumstances. It 
would be difficult to give a rule as to the kind of motion in a given case, for an 
alteration in the relative sizes of the solids concerned wall totally alter the character 
of the motion. 
As an illustration of this take the case of two parallel planes of finite breadth 
placed symmetrically one behind the other in a broad stream. If the second plane be 
of less than a certain width, free stream lines will proceed from the edges of the first, 
und the second will be in still water. 
Now suppose the second yfiane broadened until it cuts the stream lines from the 
first plane. 
The character of the motion is changed. Two vortices will appear behind the first 
plane, and in addition there will be free stream lines from the edge of the second 
plane. 
No method has yet been discovered which will give solutions of cases where there 
is motion on both sides of a surface of discontinuity. In the problems treated in the 
present paper there is always still water on one side of a free stream line. 
In the present section the motion considered is in two dimensions, the boundaries 
are ])lane, and the free stream lines are non-reentrant. 
Let X, y be the coordinates of a point in the liquid, (f>, xfj the potential and stream 
functions respectively. The region in the iv plane corresponding to moving liquid 
in the 2 plane will be bounded by straight lines rp, infinite in one direction at least 
and parallel to xp = 0. 
The area in the w plane is therefore of the nature treated in Problem I. (a), that 
is, it is bounded by a polygon whose angles are alternately four right angles and 
zero. 
This area, then, by means of Problem I. (a), may be transformed into the part of a 
new ti plane in which q is positive, where u = j) + iq. 
In this u plane the boundaries of the liquid, both the plane boundaries and the 
free stream lines, are represented by the line q = 0. 
Let, as before. 
V = log ^ 
aw 
dio' 
+ 
We have seen that V is a potential function, considered as a function of ip, and, 
therefore, it is also a potential function considered as a function of p, (/, for (p, xp are 
conjugate with respect to p, q. 
Further, we have seen that along a straight boundary \p = constant we have 
clYjdxp = 0 , and, since all the straight boundaries correspond to portions of <7 = 0, we 
