MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
409 
Between u ■= a and u ~ 1 we have 
dp 
- a^) {v/(l -«')+ 1 -P"+ y(l + v/l - Cp)], 
therefore the length of the pipe is 
(1+^1^) f‘ . „ - f' —+(1+ vT^-) r A 
'' ‘ ^ ' P. pW{ p)-^ - a~) Us/iir — a^) ^ ^ ''J«a/(A" - «") 
Now 
dp 
■' « P“ \/ {P~ — "■) 
£ 
dp 
Vi / - «') 
log 
2 [sin 
1 + V(^ ~'d) 
(■p z= a sec 6 ) — - ^ .^(1 — a^) 
cosh 
-1 
W(}-P^)dp -r / ^ / N 
, - - - / --- o -TV = B {a) (say), 
up-Vip-^ — «”) 
an elliptic integral which will be V)rought to the standard form below. 
Therefore the length of the pipe is 
where 
From 
we have 
( 1 + ft) & 1 + ^ , 7 \ T / \ 
~ log ■ + (1 + ^) L («), 
6 = v/(l — 0^^)- 
U = \ to U = CO 
£ r'r = (1 + ^ # = (1 + 0 M (a) (say). 
where M is another elliptic integral to be reduced below. 
Summing up, the breadth of jet is tt ; that of the aperture is 
77 4 - 2 (1 + v/l - cd) M (a), 
so that the contraction ratio is 
TT 
77 + 2 (1 + \/l — cd) M (a) 
where the length of the pipe is 
] + x/(l - cd) 
\/(l - - log 
1 + va -«-) 
4 - (1 -j- v /1 — L (ft). 
3 G 
MDCCCXC.-A. 
