412 MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
The only singular point for dzichv is at u = 0, and, therefore, from Problem II. (c), 
dz /y/w + ly 
dvj \\/u — 1 / 
dz 1 f\/u-\- ly 
dn u ~ a \\/u — 1/ 
Along the free stream line 
ck __ 1 A + 1 + 2iy/(-p} \i 
djy 2^ — a \ y — 1 / 
where p lies between 0 and 
If 
we have 
Therefore 
' 00 . 
Y — tan ifj 
dx 
2^ + 1 
p-\ 
P-1 
— cos 2i// 
sin 2x}j 
p — — cot^i//. 
dx cos"-\jr cosec^-v/r 
d-v/f 
dp 
dp 
= 2 
cod'p + a 
cosp cosee^p 
cnt~p + a 
If we put cos p = k ill the first we get 
X 
= -dr^ 
N2 a + (1 - a) 
-cIX, 
dx 
1 - X 
- — 2 
17.1 
dx 
a + (1 — a) X^ 
1 1 — A, \/<l 1 
— 2 log -- — 2 -—- • 
® 1 + A. 1 — 2a 
y(i _ „) 7^ '/(i - “) + c. 
and, in the second, putting 
sin p = p 
dfju 
(1 -«)A- 
_ 1 1 1 - \/(l - a) 
1 +^{1 - a)p. 
+ TT. 
we have 
