414 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
Case II .—Impact of a stream against a plane. 
A stream of given breadth impinges at a given angle against a plane. 
The impinging stream is bounded by the stream lines xjj = 0, xjj = n. 
The stream line which branches at the point B on the plane is i// = tt. 
The diagram in the lo plane consists of two infinite straight lines with a semi¬ 
infinite one between them, as in the figure. 
= 
Y = m 
Y = 0. 
In transforming to the u plane we suppose that </> = — co corresponds to i oo, 
and that w, = — 1, = 1 are the extremities of the plane. 
We must then take u = a, an unknown constant for the point B, where the stream 
line xfj = TST divides, n.nd observe that a < 1 > — 1. 
We then have 
dw _ . u — a 
d,u (n — 1 ) (u + 1 ) 
where 
ttA 
ttl 
1 + a 
1 
TTT 
1 — « 
= TT — ZS 
J 
therefore 
A 1 J 2w 
A = — 1 and a = -1. 
TT 
Along g = 0 we have, therefore, the following arrangement of points :— 
free 
ris:id 
rid 
free 
0 = — CO “0=0 
« = — 1 0 = CT 11 = a 
u = 1 
= TT 
0 = — 00 
