MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
405 
Also 
siii“ ^ a — taii~^ 
Therefore the contractiou ratio is 
a X —1 
y(l - ^ 
\ TT d d'^ — TT'' 
Now, shotting rid of the special iiaits, let d be the breadth of the vessel as before, h 
the breadth of the aperture, and c that of the jet, then 
/.• = 
\ fd c\ , -Idc ■ 
1 + f - Jtan-i ., — . 
TT \c dj d~ — C" 
If d be very large compared with k, we get 
k = c 
. -1 d d 
1 + — Lnn. tan ^ 2 - 
TT C d 
TT + 
This is the result obtained by Rayleigh'" from Kirchhofe’s solution for the case 
of an aperture in an infinite plane bounding wall. 
As djk decreases from infinity the contraction also continually decreases, until when 
d — k the contraction is zero. 
In order to get some idea of how soon the finite breadth of the vessel aftects the 
contraction ratio perceptibly, consider the case when 
or 
2 cd 
d — [V -}- v^2) c. 
The contraction ratio is then 
1 + 
TT 
and 
TT 4 
d — (1 -f 2) |d', 
so that the finiteness of the vessel has very little effect on the jet if the breadth is 
more than twice that of the aperture. 
The equations to the free stream line x}j = tt are 
__ / _ 0, 1 I 
dj) — 
dy 
dp 
1 v/(/-l) 
T - «") 
* ‘ Phi]. Mag.,’ Dec., 1876. 
