MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
417 
The diagram in the iv plane will then consist of one infinite straight line and one 
semi-infinite, as in the figure. 
= TT 
Y' = 0 
We suppose xjj = 0, xjj = n to be the bounding stream lines. In transforming to 
the u plane we take the point \jj = 0, tf) = — oo tobew = — oo; xjj = 0, (f) = co to 
be u = ct, and then use the two arbitrary constants at our disposal by making the 
edges of the aperture to be u = — h, u = b, and the branch point of i// = tt to be 
M = 1 -f- a, where 1 + « > ^ > 
We then have 
dtv . u — a — 1 
— = A-, 
cm u — a 
and, remembering that the bounding stream lines are xjj — 0, xjj = tt, we obtain A = 1, 
so that 
cho — cc — 1 
chi u — a 
There is only one singular point for dzjdiv, viz., u — a1 , and at this point 
k/tt — 1 = 0 , so that no factor u — a — 1 appears in dzjdu. 
The arrangement of points on = 0 is as in the figure 
rigid free rigid 
——— -1-I---I-X-— 
■^■ = 0 u = — h u a ^ = TT u = h = \J/- — vr 
and hence 
_ (a + l)u — + \/(a + ly — (u^ — h“) 
dw h{u — a — 1) 
so that 
_ {a + 1) tt - 5"“ + v/(ft + 1)=^ - h~ y(^r - b”) _ 
clu b {li — a) 
When w = i 00 , we have 
dz _ + 1 + 4- ip _ 
dw b 
and, therefore, the velocity of the stream is 
_ b _ 
(a -F 1) -f \/(ci + 1)2 — 62 
3 H 
MDCCCXC.—A. 
