418 
MR. J. H. MIOHELL OR THE THEORY OF FREE STREAM LINES. 
When u — a, that is at a great distance along the jet, 
dz a {a + 1) — -\- i \/ {a 1)^ — Id's/ (Ir — cd) 
dw b ’ 
so that the inclination of the jet to the wall is ultimately 
, 6^ — a (a + 1) 
COS~-^-r-- 5 
b 
the final breadth of the jet being tt. 
There appear to be two constants here, whereas there ought only to be one. 
The explanation is that we have not yet expressed that the two parts of the 
boundary are in the same plane. This will give a relation between a and 6. 
Between u = — h and u — a we have 
Now 
and 
dx _ 1 (a + 1) p — 
rfp b p — a 
dy _ \/ (a -f s/jb"- — p~) 
dp b p — a 
c,y(d -f) 
1 
1 
1 
1 
V{b^~-f) 
U/ fJ 
p — (t 
\(p- 
dp 
a) ^(Ir - 
dr 
J \/(&^ — cd) r~ + ‘2ar — 1 
dr 
where p — a = 
\/b~ — a? r + 
W 
1 
- fd) 
1 
log 
\/(&”“ - a") 
(Id — a^) J d — cd 
V ~ ’’ + ^(52 _ r” + 2 ar 
d — ap + \/~{W — (d){W — p‘^) 
— 1 
a — p) 
Thus along ^ = 0, measuring from the edge A, we have 
X — Xx = 
y 
rt. + 1 , , J. , a (« + 1) — , a — p 
b ^ a + b 
\/(a + H) - 
— — Cl') log 
— ap -{- \/{b~ — a^) (6" — p^) 
{a — p) b 
+ a sin-1 _ ^ T 
