416 
MR. J. H. MICHELL OR THE THEORY OF FREE STREAM LINES. 
therefore 
log - (1 -i-p) ~ log (1 -p} + A' I 
2/ = v/(l - a^) log - 1 - p)-\- Tff 
where ^3 lies between — 1 and — go. 
1 
cos 0 
If we put p = — ^ . in the first and p — — ——- in the second, we net for xh 
1 1 f) COS ^ ’ & r 
0 0 ~1 
a’l = (1 -j- «) log cos - — (1 — a) log sin - — a log cos ^ I 
= 77 
Vl = \/(l - log cot + 77 - OT, 
and for xjj = 0 
X.-, 
(1 + a) log sin -“(!—«) log cos .a — log cos 6 A' 
A .... 
I 
J 
where in both cases 6 lies between 0 and 4 tt. When 0 = 4 77 in both we net 
2/3 = \/(l “ «") log cot 1 - - (9 + 717 
therefore 
Xy — x,2 — A — A' 
2/1 — ^2 = — 277 
A A/ 7r+(77— 2 ct)« 
A — A = - ... _ r = 77 W( 1 — or), 
and the ecjuations of the bounding stream lines are now completely determined. 
Case III .—Flow of a broad slreani past a plane wall in which there is an aperture. 
Let BC be the aperture in a plane wall ABCD, and let the stream flow from left to 
Tlie left boundary of the issuing jet will be the continuation of the stream line AB. 
The right boundary will be one branch of a stream line u Inch divides on the plane 
CD at some point E. 
