MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
423 
The former gives 
\ ~ ah 
X — x^— — 
1 h —p 
-locr - 
c — b ^ 5 + 1 
(XC ““ 1 G X) 
log ^ ^ between p = — 1 and p 
c — h 
= h. 
and 
ah — 1 V — h ac — 1 , c — , 
X — x-p,— - log , -7- log -r between p ■= 0 and p = 1 . 
G — h ° 1 — b c — b ° c — 1 ^ 
Also in passing the point u = b, x changes by 
- 1 ) ^(1 - 5 "-) 
TT-;-- 
therefore, putting p = h — e in x — Xj^, p = 6 + e in a: — x^, and proceeding to the 
limit, we get as the breadth of the aperture, 
1 — «5 
c — h 
log 
1 + h ac- 1 . c + 1 , - 1) y(l - ¥) 
-t 7 “1“ 7 tog’ ~ -t- TT ; 
1 — 6 c — b ^ c — 1 ' c — b 
The direction of the jet makes an angle 
cos ^ 
with the bounding planes. The breadth of the jet is 
To sum up : Let d be the breadth of the pipe, h of the aperture, I of the jet. Let, 
further, v-^ be the velocity at ^ = — qo , that at (^ = qo , Vg that of the jet. 
- — a + — 1) 
and therefore 
Vi c — h 
^2 c — a 
■L\ _ c — 6 
— Vj a — b 
— ^3 
I v^-v^ 
h 
e — b ri — a6, 1 + 6 ac — 1, 1 + c 
~ 7 r{a - 6) I V^~h 1-6 c - 6 -1 + c 
+ 77 
- 1) y(l - 6«) 
c — 6 
with 
( 1 ) 
( 2 ) 
( 3 ) 
( 4 ) 
( 5 ) 
(c ~ h) {a + — 1) = «c — 1 + \/{cd — 1) ■— l) , , , (6) 
