400 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
and 
Therefore 
U + = 2 i log ~ - 
dz -iiz\f{u)du 
(ho ~ ^ ■ 
Now we have obtained the transformation from the w to the u plane in the form 
[Problem 1 . (a)]. 
Therefore 
clz dz (ho . . - il'2.\flu)du 
which gives 2: as a function of u. 
This is the general solution of the hydrodynamical problem before us. 
Near an angle of the boundary or a branching of a stream line we shall have 
dzjdu = A {u — Uq)"'. 
The determination of the index n rests on principles already used. 
If the internal angle of the boundary, or the angle between the two branches of tlie 
stream line be a, then 
u = - — 1 
TT 
For ex:ample, if a stream line divide on a plane wall, n = 0, and the point of division 
is not a singular point for dzjdu, although it is for dzjdiv. We may then lay down the 
following rule :— 
Near a singular point 
dz . , \ '‘^1 
.7.7, = ^'0) - 
where a is the internal angle of the boundary except when this point is a point 
of branching of a stream line, in which case 
= A 
for at that point 
^ ^ ^*0) - 
and 
dwjdu = B (t 7 ““ ^0). [Problem I. (a)]. 
We shall now go on to consider such cases as are susceptible of tolerably simple 
treatment. 
We shall suppose there are only two free stream lines, and throughout take the 
velocity along them to be 1, so that V = 0. 
