MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
415 
The only singular point of dzldiv is at u = a. 
Hence 
clz _ 1 — cut + (1 — «-) .y/(l — u^) 
dio u — a 
and, therefore, 
dz _ 1 — ail, + ^(1 — fd) y/(1 — \d‘') 
du {u — 1) (u + 1) 
+ \/(l — cd) 
1 
\/(l - «“)' 
When u = ± 00 , we have 
dx 
dy 
Therefore, if 9 be the inclination of the stream to the plane, we have 
^(1 - fd) 
tan 
9 = 
or cos 9 — a. 
Since we have before found 
this gives 
a = 
2 c 7 
TT 
- 1, 
77 COS 9 = 2c7 — 77. 
This equation merely expresses that the momentum parallel to the plane is 
unaltered by impact. 
Along the stream line <// = 77 we have 
therefore 
dx _ 1 — ap 
dp — 1 
dy 
dp 
v/(l - 
\/(P" - 1) 
1 
I 
;> 
® = 2 (P® - 1) + 5 ^ 
= log (p + 1) - log (p - 1) + A [ 
y = v/(l - cd) log (p d- v/p‘^ — 1 ) 7T - m. 
where p lies between + 1 and 00 ; and for \p = 0 we have 
dx _ 1 — ap dy _ ^, ,, 1 
dp p^ — 1 dp ^ ^ V {p^ 1) 
