56 Proceedings of the Royal Irish Academy. 



and, for a single vortex round which the circulation is n, 



ifx ^ sin (7rA"'(^- (I -1/3)1 /rKx 



For a number of sources or vortices ir is formed by addition of terms such 

 as (56) and (57). 



The elimination of Z between the (»•, t) transformation formed in this 

 manner, and that (s, 2) ti'ansformation which represents the relevant region 

 of the z plane upon an endless series of strips in the positive half-plane of Z,, 

 gives a relation between z and w which specifies flow with the prescribed 

 sources and vortices and the known or prescribed boundary. 



For example, it is known that the transformation 



r = {iXI-2n) log {zla) (58) 



represents the region outside the circle of radius a in the z plane, whose 

 centre is at the origin, upon the infinite series of semi-infinite strips of 

 width A in the half-plane on the positive side of the real axis of Z,- Hence 

 this relation, combined with (57), defines a relation between w and 2 corre- 

 sponding to a vortex in presence of a circular internal boundary, or to n line 

 charge in presence of a circular conductor. The resiilt of eliminating Z is 



in , sin ttX ' I {iX/2Tr) log z/a) - a - iB\ 



— iotr ' ' — • 



2v . ° sin 7r\-' I (a/27r) log (z/a) - « + i/3) 



(59) 



It is easy to verify that this corresponds to the familiar formula 



/r= (i/i/27r)log|(3- 2,)/(::-s,)|, (60) 



where s, and S; are image points with respect to the circle. 



A single vorte.x in presence of an elliptic boundary is represented by 

 formula (57) in combination with 



s = ccosh|« -(27ri/A)^i; (61) 



and a single line-charge in presence of a prismatic conductor is represented by 

 formula (57j in combination with 



dzld,Z = KS\ (sin ttX"' (^ - 7^ | ^ " "^ ^ (62) 



the parameters being subject to the conditions explained in Article 8. 



These examples have not allowed for a circulation round, or total charge 

 upon, the boundary itself, but this is easily provided for by introducing 

 a linear term into the (t. Z) transformation. The general form of this 



