J[R. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
393 
and on tills semicircle let 
10 — = Ji (cos d i sin $), 
so that 
ilz > , 
, = AR"' (cos Hr 6 i sill n,.6). 
dw ^ ' 
to 
Consequently, as we pass from 6 = w to 6 = 0, dz/dw goes from 
AR"' (cos n,.7r + i sin Urtr) 
AR'". 
The amplitude of dzjdw has, therefore, decreased by Urir. 
But the increase of amplitude is tt — ar where is the internal angle of the 
polygon which corresponds to 
Therefore, 
SO that the transforpiation becomes 
which is the formula given by Bchwarz and Christoffel. 
Problem I.— Special Case, 
For the study of non-reentrant free stream lines we require a special case of this 
formula. 
Suppose the polygon to consist of a series of straight line sinfinite in one direction, 
all parallel to y = Oj so that the angles of the polygon are either 0 or 2tt, and, there¬ 
fore, n,- = + 1 or — 1. 
Let <^i,, correspond to an angle 2?r, and, therefore, to an end of a line within a finite 
distance of the origin, and let (/> 2 ,. correspond to an angle 0, and, therefore, to the 
adjacent ends of two lines at an infinite distance from the origin. 
Schwarz’s formula then becomes 
dz 
dto 
= ita 
w ~ 01 ,. 
3 E 
MDCCCXC.—A, 
