MR. J. H. MICHELL OX THE THEORY OF FREE STREAM LINES. 
407 
therefore 
<]z i 2 — — 'll" 4- 2^(]. — — v^) 
du 'll 'id — cd 
When u = 0, that is, at a distance upwards from the tube, 
^ (2 - a- - 2 . 
div cr ' 
therefore the breadth of the vessel is 
77 -«^ 
2 - cd -2 v^(l - cd) 
In passing over u = — a, x increases l^y 
2"(1 - a ^) 
ir ' 
therefore the breadth of the tube is 
llCl- 
Ttt 
2-od - 2^/(1 - cd) id 
Hence 
therefore 
- - ^ (1 - aH = - [Sa^ _ 2 + 2 y(l - ci^)} = h (say). 
d. — /t + Ttt 1 
Ttt cd 
Substituting in the value of d we have 
d 
d - k + Ttt / 
+ TT = TT + Y 
d - 7 ; 
d — k -]r Ttt 
or 
— /i + Itt/ 
f/ -|- h — 277 = y7 \ [d — k) [d — Z; + 
On squaring this gives 
dk — 2 ' 7 Td “h 77 ^ = 0, 
or, getting rid of the special unit of length, if c be the breadth of the jet, we have 
dk — 2 cd c® = 0 , 
or 
{d-cf=d{d-k). 
When d is very great we get 
and the ratio of contraction is 
k = 2c, 
