1891.] Fluid Motions in favo dimensions. 187 



and since the z boundary has no singularity at a = 



Aic 2 _ 



Q^ Q 2 Q^ 



whence — 5 — = J(l — a 2 ) or c 2 = = ttz — —^- (13). 



c 2 v 1 + V(l - » ) 



Also £1 = + i— when w = l so that B = i, and the relation be- 

 tween z and u is 



1 - au + V(l - a 2 ) V(l - w 2 ) 1 + au + V(l - a 2 ) V(l - -m 8 ) ' ' 

 u — a u + a 



Now remembering the identity, 



V( l + a) \/(l + u) + V(l - a) V(l - u) 



we have 



= {1 + au + V(l - a 2 ) V(l - '^ 2 )} 2 > 



* - * [i ■+ V (i - o] V(1 -"?+^r J) (i4). 



du 2 L n *J{u> - a 2 ) ; v 7 



This corresponds to the case of liquid flowing directly against 

 a disc with an elevated rim, the rim being the line in the z plane 

 that joins the points corresponding to u = a and u = 1. Observe 

 by way of verification that if we make a = 1 there will be no rim 

 and we get the right transformation for fluid flowing against a 

 plane disc, viz.: 



! = -l[l + V(W)] do). 



8. Suppose now that a is very nearly 1 and take 



l-a 2 = k\ d z = l-k 2 = k' 2 (16), 



where k is small. Then equation (14) may be written 



dz_i r V(l-Q , ,, V(l-a 2 ) 1-u* 



du~2 y{u 2 - a 2 ) [ X + Vii a ;i + V(w 2 - « 2 ) VO 2 - a 2 )} ' 

 and the height of the rim is 



1 r \f-y) (1 + V(1 _ „*» + f^L + a^jI 



du 

 ..(17). 



