MR. J. H. MICHELL OR THE THEORY OP FREE STREAM LINES. 
411 
while the aperture becomes 
A; = TT + 2 (1 H- ^/l - a^) 
= 77 + 2 + 2 v/(l - a3) 
1 - "2 log ”/ 7 i-^ H" ^ 
«7 ® +(1 — fl.) 
to the first order, and the breadth of jet is c = tt. 
So that 
I 77 + 4 , g. 
(1 - 
and 
therefore 
k 4 (tt + 2 ) 
Ic 77 + 2V 77+2"^ 
i /i - ^ 
Z ; 77 + 2 y 77 + 
a- 
77 + 2 Z 
77 + 4/7 
which is the approximate expression required. 
Example 7 F,— Flow from an aperture in a pipe in which the water is at rest. 
By considering the symmetry of the motion this can be reduced to the case of flow 
from a rectangular vessel in which the aperture extends from the bottom up, and the 
bottom is continued into a horizontal plane, as in the figure. 
The IV diagram consists of two doubly infinite straight lines, xjj = 0, i// = 77 , as 
before. 
But if in the u plane we make u = —co correspond to ^ = 00 , the arrangement 
of points is not the same as before and the notation must be altered. 
We take, then, w = 0 for the edge of the aperture, u = 1 the right angle of the 
vessel, and then put u = a for cf) = — 00 . 
We then have 
dw 1 
dll u — a 
and the arrangement of points on O' = 0 is as in the figure. 
rig-id rigid rigifl 
-1-1 - --1- 
= 0 — ^7 -if = ci y = 0 M = 1 
3 G 2 
free 
Y^ = 77 
