44 Proceedings of the Royal Irish Academy. 



together with the tangents at its extremities. A curve-factor may be called 

 improper if the locus correspouding to B real has angles anywhere, the case of 

 angles at the points wheie the curve joins the two straight lines being a 

 particular ease. 



Now on reference to fig. 2 it is seen that the locus B real corresponds to 6 

 moving along the whole of its real axis with semi-circular infinitesimal detours 

 round the branch points of (r' {0), and in particular such branch points as there 

 may be at 6 = ± c. On the other hand, the locus v: real corresponds to 6 

 moving along the part of the real axis between - c and c, making complete 

 circular detours round the branch points at = ± c. If a semi-circular detour 

 round one of these points makes no abrupt change in the vector-angle of C, 

 then a complete cu'cular detour will make no change ; so if the locus 6 real is 

 smooth at such a point, so also is the locus w real. An angle in the locus B 

 real at c would be represented by an angle twice as great in the locus v: real. 



7. Pei-iodic Cwner-factors. — In the case of a conformal representation in 



which dzjdw is periodic, the presence of a corner at the point corresponding 



to w = a on the closed or periodic curve in the z plane defined by -^ = implies 



a similar corner at the points w = a ± n\, when n takes all integral values. 



Therefore the diflerential form of the (z, w) transformation includes, instead 



of a single Sehwarziau factor or power of v: - a, a, corresponding power of the 



infinite product 



{w - a) n {(w - aY - n-y), 



or rather, since this form is divergent and constant factors do not affect the 

 geometrical propertj', 



But this is 



sin {■jriiv - n)/Xj, 



and so it appears that the periodic corner-factor of linear period X is 



sin JTT {vj - u)j\\, 



the power to be employed depending upon the angle at the corner in the same 

 manner as in the case of Schwarzian factors. 



B3' way of confirming tliis statement it may be observed that, as vj passes 

 through real values from + co to - co , the vector-angle of 



[sin|7r(w-a)/Aj]'' 



increases abruptly by ^tt iu passing any such value as a + n\, but remains 

 constant between a consecutive pair of such critical values. Thus if dz/div 

 be equated to a product of such factors, the curve in the s plane which cor- 

 responds to i|r = is a succession of straight lines interrupted periodically by 



