MR. J. H. MIOHELL ON THE THEORY OF FREE STREAM LINES. 
419 
In like manner between u — a and m = 6 on v/; = tt, we have 
a + I , , a (a + 1) — p — « 
--log 
y-yB = 
\/{a A 1)2 - 62 
lo 
b - a 
W — ap + \/(62 - a°)(b^ ~ P^) 
ip — a) h 
— a sin-^-^ + cif'Y + \/(^^ — P^) 
To find £Ca — and — 2 /b we must put p — a — e and p = n -j- e respectively in 
the above, and then decrease e indefinitely. 
If we then remember that, in passing the point p = a, x + iy increases by 
ITT 
we get 
[a 4~ 1) a — IP + i y/(« + 1)^ — P \,/{lP — a~) 
a {a + 1) — b~ — a tt 
6 + « h 
x^-x^ = 2 {a + 1) + " -^log + ~ x/{a + 1)^ -- h- 
- a~) 
and 
_ \P (cc H" 1)" - b“ I ^ r 1 rt j , T \ 
.Vb — lu — — TTCI^ --d- - {h-^ — n (a + 1) 
or 
In the case we are considering — yp^, and, therefore, 
IP — tt (a + 1) = a 
IP = (tt d- 1)2 — 1, 
so that 
(tt + 1)2 — 52 = 1 
tt (tt + 1) — 62 = — tt. 
Using these equalities, we now get for the breadth of the aperture 
— a^A — 2 (tt + 1 ) + - log + TT —p- 
where 6/(a + 2) is the velocity of the stream, rr the breadth of the jet, and co^~^{ajh) 
the inclination of the jet to the plane. 
To get rid of the special units let Vt^ be the velocity of the stream, i’g of the jet, h 
the breadth of the aperture, c that of the jet. 
Then 
, 0 + 2 
3 H 2 
V, 
