Leathkm — Periodic Conformal Curve-Factors and Corner -Factors. 55 



factor of the tj'pe of (s-.«, except the constant term which leads to ^5,. The 

 product of all these curve-factors, if it is convergent, is the curve-factor 

 corresponding to /'(")• 



It may be noticed, in passing, that the form of /'(«') suggested by 

 formula (37), as it has branch-points, does not admit of the method of 

 deriving the corresponding curve-factor which has just been described. 



13. Tlie specification of fields with logarithmic singularities in th£ region 

 outside a cylindrical or prismatic houndary. When it is desired, particularly 

 with a view to physical applications, to in^■estigate fields with assigned 

 logarithmic singularities, such as sources, vortices, electrodes, or line electric 

 charges, in doubly connected regions of the kind under consideration, 

 a simple formulation is available. The procedure is simply to employ 

 periodic curve-factors and corner- factors, or any other available method, 

 to represent the doubly connected region in the z plane conformally and 

 repeatedly on an infinite succession of semi-infinite strips in the principal 

 half- plane of a variable c,; when this has been done, sources or vortices 

 (in the hydrodynamical application) may be taken account of in a {w, Z,) 

 transformation, each source or vortex at a point a + i^ in the K plane 

 being accompanied by such an image at the point C = a - ;/3 as is required 

 to maintain the constancy of \p along the axis of Z, real. But a single source 

 or vortex at ^ = ^^ = a + i/3 is not, in the circumstances, a representation of 

 a physical possibility ; what is wanted is an endless series of sources or 

 vortices, all similar to one another, at the points Z = Zu + nX, where n takes 

 all integral values, balanced by the corresponding series of images. This is 

 the only way of ensuring that every strip in the Z plane which corresponds 

 to the complete field in the z plane is equipped with a singularity 

 representative of the single source or vortex which is present in that field. 



The w which, in the absence of a boundary, would correspond to such 

 a periodic singularity, would be proportional to 



logiiZ - t.) n[{Z ' Zof - n^x^n, 



or (as in article 7 above) to 



logsin(7rA-(^-2o)!. ' (55) 



When 1// is to be zero for Z real, a corresponding term involving the complex 

 conjugate to Zo must be included. 



Thus, for a single source at Z = » + ifi, which produces liquid at the 

 rate m, the form of ^'J is 



w = - {m/2Tr) log [sin { irX-' (? - a - i/3) j sin { ttA"' f ? - a + i/Sj } ] ; (56) 



