192 



Mr Love, On the Theory of Discontinuous [May 4, 



will be the stream-line that divides, and the boundary in the 

 u plane is 



-1 



The boundary in the fl plane is easily seen to be 

 1 0=0 



a 



-i 



This O-region can be conformably represented upon the half-plane 

 u by means of the relation 



(21), 



dO L _ A 



du \f{u 2 —l)u — a 



Ai 



whence O = log 



l-aw+V(l-a 2 )V(l-w 2 )' 



so that -7- 

 du 



u — a 

 l-au + y/(l-a?)^(l-u*) 



v/(l-a 2 ) 



in which the index — 



(u — b) (u — c) 

 Ai 



•(22), 



has been determined to be unity by 



V(l-0 



considering that there is no singularity in the z boundary at the 

 point corresponding to u = a. 



10. We may now suppose that the lamina is part of the axis 

 of %. If we take u to lie between — 1 and 1 and integrate (22) 

 between these limits we shall get an expression for the breadth of 

 the lamina. 



Writing the equation (22) in the form 



dz ab — 1 1 ac — 1 1 

 du~ c—bu — b c — b c — u 



+ 



V(l-a 2 ) V(l-a 2 ) 

 V(1-m 2 ) c-b 



b 2 -l 



+ 



c 2 -l 



_(u - b) V(l - u 2 ) (0 - u) V(l - u\ 



