422 MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES, 
and therefore 
A = — 1 
TTJ 
b - 
a 
TV 
b- 
c 
TT — V7 
c — 
TT 
The arrangement of points on g' = 0 is as in the figure 
rigid free free 
1/^=0 
W =—1 Y ^=0 W = 1 
rigid 
— I- 
u—a 
rigid 
u — r. y: 
and from Problem II. (a) we have 
so 
that 
dz _ au — 1 + \/{ci~ — 1) — ]) 
d.w u — a ’ 
dz _ au — 1 + y /ffd — 1) y/A" — 1) 
du (li — h) (u ~ c) 
Hence, in the pipe, the velocity at — oo is 
1 
TTuT-T) = “ - - ')■ 
and at ^ = CO is 
c — a 
ac 
'.c — I + Y/(fr — 1) y/(c2 — 1) 
The breadth of the pipe is, therefore, 
/ . /-I-^ ^ ^ - 1 + ^/(rd - 1) y/(c2 - 1) 
7r(a + va^ — 1) = (tt — ct)- 
c — a 
Now 
therefore 
77 — ZS - 77 
c — a 
c — b 
(c — h){a-\- \/a^ — 1) = ac — I + — 1) -v/(c^ 1). 
This is the relation between a, h, and c. To find the size of the aperture we must 
integrate between u= — 1 and u = l. Within these limits we have 
dx 
1 — ap 
1 — ab 1 ^ ac — 1 1 
dp (b — p) (c — p) c — b b — p c — b c — p 
djl _// 3 _ >, \/(l - f) _ _ v/(«" - 1) 
dp ' ' {h p) {c — p) c — & 
s/1 — jp^ 
1 — ah ac — 1 
h — p) c — p 
c ^ b 
