408 
MR. J. H. MICHELL OH THE THEORY OF FREE STREAM LINES. 
'I'his particular case was the first solution of free stream lines given, and bv 
II ELMHOLTZ. 
Lord Rayleioh'" has given tlie equ dion 
I 
(/ 
for the coefficient of contraction when there is a tube projecting inwards in a vessel of 
finite breadth. 
The assumption mabe is, however, not that of this example, for he has taken the 
velocity along the bottom of the vessel to l)e the same as that at a distance upwards 
from the tube. 
This is one of the very few cases in which the contraction can be determined 
accurately from elementary pi'inciples. 
It is scarcely worth while to put the proof of this down, but it is worth remarking 
that the corresponding case in three dimensions, can also be worked out, viz., the 
case of one circular cylinder projecting far into another. 
Let 7q, be the radii of the cylinders iq > r.,, and 7q the radius of the jet, then 
rpry — = 0. 
This includes the case first noticed, I believe, by Borda,I of a cylindrical tube in 
an infinite plane wall, the proof being just the same. 
Example III .—Tube projecting into a vessel of great breadth. 
If we take the case of a tube projecting into a vessel of great breadth, we do not 
get such simple expressions. 
The transformation to the u plane is the same as before, but there is now a singular 
point at u — 0, corresponding to (f) = — co . We arrive at the correct result by first 
supposing the vessel of finite breadth, so that there are points at n = — a, u — — 1 ), 
u = h, u = a, for each of which a/?? ~ 1 — — b, and then making b vanish. 
We have, therefore. 
and 
, — /, o 1 \/( i — "') “k \/( 
//v p{i/~ — a~) 
7“ = 1 \/( 1 — r I — r (I + Cl «') \/( ^ '<');■ 
ait n- \/{u- ~ n~) ■ 
* “ The Contracted Vein," ‘ Phil. Mag.,’ Dec. 1876. 
t ‘ Mera. de I’Acad.,’ 1766. Paris. I o\re ihis reference to the kindness of Loi'd R.wlkigH. 
