MR. .T. H. MICHELL ON THE THEORY OE FREE STREAM LINES. 
Similarly the other piece of the bottom of the vessel is of length 
Ap + r, + C x/Tl - /r) 
j _ n Ap + l’> + 0\/(l - //) ^ 
J 4 (p — c) v' Ip — a) {p — b) 
When (f) = — ao, that is, when u = c 
dz _ . Ac + B + C — C-) 
dw ' \/{c — a) (h — c) 
Therefore the velocity in the vessel at a distance from the aperture is 
\/{c — a) {b — c) ^ 
Ac -}■ J3 + 0 (1 — c"^ 
and therefore the breadth of the vessel is 
Ac + B + C \/ (1 — c®) j , 
^ - ,.) it -.) = ("‘V)- 
The breadth of the aperture is 
d — ly — I.,, 
and the breadth of the jet is ultimately it. 
The question is now reduced to a matter of the integration of /j, 4 - 
For the general case elliptic integrals occur, and the expressions for our purpose 
may as well be left in their present form. If, however, the aperture be in the centre 
of the vessel, the integrals will work out, and we get a simple expression for the 
contraction of the jet. To this we now proceed. 
Suh-Exaraple I .—Jet from an ajjerture in the centre of the bottom of a rectangular 
vessel. 
In this case we may take the angles of the vessel at u = — a, u — a, a < 1 . 
The expression for dzjdu then reduces to 
We now have 
Now 
dz ^ 1 v/(l - cd) + v/(l - ,d) 
dll a \/ id — 
1 1 v/(i - «“) + W(i ” r) 
— 0 ? 
dX 
dp 
- v/(l ~ a~\~) 
X = 
p 
sin ^ aX 
L ’ 
-la 
— Sin ' a 
3 F 2 
