394 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
It is plain that there cannot be more than one factor more in the numerator than 
in the denominator, so that we can write 
dz 
dio 
= Aw + B + 
) 
and, therefore, on integrating 
z = I Aiv^ + Bic -f-1) + log {w — </).>,.), 
we may at once determine the distances between consecutive lines in terms of the <f>r, 
or in terms of the D,.. 
For consider the passage of w around the small semicircle R above described. 
We have 
IV — (l)r— Re'®, 
therefore 
R " d (Re'®), 
= iQ,. [ tW, 
* TT 
= — nrGr- 
So that y increases by — in passing the point <^0,) therefore, the distance 
between the parallel lines r and r + 1 is — ttC,-, or, in terms of the </>,-, is 
— ttAII 
4>r ~ (fi-Zf 
After we have fixed on the angle which is to correspond to (^ = i 00 , we can in 
general choose the position of two other points and then the transformation- 
Ibrmula is determined to an additive constant. 
For example, take the case of two doubly Infinite lines AB, CD, with a semi- 
infinite line EF between them. 
C 
H 
E 
F 
A 
B 
We may take the zero angle (A, C) to be ^ = Aoo , and the angles (B. F) (F, D) 
to be (/) = — I, (^ = -p 1. Then if the angle E is ^ = c, we have 
dz A(w — c) 
dio {w — 1) (w +1) 
