402 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
Examjjle I .—A rectangular vessel of given width has an aperture in the bottom. 
Here there are only two angles, each of which = i 77 , and therefore I — clJtt — I. 
Let tlie two angles be w = a, u = h, where l>h>c>a>— 1. 
Then, observing that 
(1 — au -{■ \/l — \/1 — = ) \/ 2 (1 — ^ u \/^ [I a) \/1 — uY 
we get 
__ 1 — ft)\/l + ?6 + \/^(l + ft)\/l—'U}{\/^(1 — />)\/l + a + v/^(l+ h)\/l —u] 
du u — c 
or 
Where 
^( u — a . u — h) 
(h _ A?^ + B + C vAl - ?r) 
dt(, (u — c) >/(u — a) . (;<■ — h) 
A = \/:j (1 — 1 1 “h > 
B = v/:j: (1 — n) (1 — 6) + \/^ (t + «) (1 + ^), 
CJ = s/i ( 1 + «) ( 1 — “h ^ 1 • 
Between u= — 1 and u = a, that is along the bottom of the vessel from the edge 
of the aperture to the angle on the right, we have 
dx _ A p + r. + C v/( l - 
dp (p — c) \/{p — a) Qj» — />) ’ 
therefore the distance between the two points is 
B A27 + r> + C ^/(l - 
J-i (c - p) \/ iP - n) {P ^ 
