Leathem — Periodic Conformal Curve- Factors and Corner -Factors. 57 

 transformation is 



w 



= ^2: + 2 -^ log 



^. , ,sin^(?-«-^^)^ 



:27r 



sin — - f?- a + i/3) 



- 2 ^- log } siu y (^ - a' - ^J3') sin y (? - «' + i/S') j , (63) 



and terms representing doublets could be introduced if desirable. 



The formula (63), coupled with a {z, Z,) transformation, whether of the 

 type z =f(Z) or of the type dz = ff(Z) dl!„ gives the specification of the 

 field with assigned singularities in the region bounded internally in accord- 

 ance with the latter transformation. The only further steps requisite for 

 explicit formulation are (possibly) integration, and (certainly) elimination, 

 and adjustment of parameters. The inner boundary may be a rectilineal 

 polygon, in which case {p{^) is a product of corner-factors in ^, or it may be a 

 smooth curve, in which case (o{Z,) may be one of the periodic curve-factors 

 considered above, or/(2^) may have one of the forms which have been shown 

 above to be suitable. 



The limitation of the method is that one cannot prescribe the boundary 

 arbitrarily and be sure of getting a solution ; one must be content with such 

 boundaries as correspond to known forms of (norf. If the range of known 

 forms of periodic curve-factors can be extended, the scope of the method will 

 be correspondingly enlarged. Meanwhile it is possible that a rough approxi- 

 mation to any particular assigned boundary might be got by a suitable choice 

 from among the focal curve-factors. 



K.I. A. PROC, VOL. XXXin., SECT. A. [9] 



