404 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES, 
and 
therefore 
p y(l - f) 
clp = — [ -— 
- — a~) ^ J (1 — 
r'^dr 
\/(l - — r~) 
= f-- { 
J a /(1 — a- — O J 
(P- = 1 - 
dr 
) J (1 - P) a/(1 — or - ?'3) 
Sill 
-1 
tan 
-1 
ar 
10 
\/(l “ \/(l O!," — ?’")j ,/(]^_a2) 
TT 1 — (C 
'1 a 
I — I — - 
^ ■2 — ' J — a 
1 — « , — (d) 
" • — 1 
- — Sin ^ a 
When u — 0 that is, at ^ = — co, we have 
dz _ _ ■ v^(l - + 1 
dtu a 
Therefore the velocity in the vessel at a distance from the aperture is 
1 + \/ (1 — Cl“) 
Therefore the breadth of the vessel is 
1 + y/ (1 — o?) _ 
-TT 
= d (say). 
So that the breadth of the aperture is 
7 7 \/( 1 ~ fd) . , 
d — " 2 L = TT + 2 - - sin ^ a . 
^ a 
Hence, since the final breadth of the jet is it, we have as the ratio of the breadth 
of the jet to that of the aperture 
TT 
TT + 
\/(l - (d) 
sin ^ a 
Now 
therefore 
and 
1 + \/ (1 — cd) _ d 
a TT ’ 
1 - \/(l - (d) _ ^ 
a ~ d’ 
2 ~ _ T 
a 77 d 
