LiiATHioM — Periodic Conformal Curve- Factors and Corner-Factors. 51 



and therefore the subject of integration in JC is, to the same degree of 

 appi'oxiraation, 



-{w - w,)"^ log sin - (w„-w,) + (i«-iy,)"i 



TT A- 



-logsin^(w„-w,) - - eot^(w„-iyi) . 



For the small circular contour, described in the negative sense, the first term 

 integrates to zero, and the second gives 



- 9.i 



P log sin ^ (w, - IV,) - ~- cot ^ (w„ - w,) 



(41) 



which is therefore the limit for a circle of vanishing radius. 



In considering the contribution made to the contour integral by the sides 

 of the rectangle which are parallel to c^ = 0, it is to be noticed that f'{w), being 

 periodic, has the same value at points on the two sides corresponding to the 

 same value of i^, no ambiguity of value being possible in the region defined by 

 the complete contour. It is also to be noticed that at such corresponding 

 points sin ir\'^(;W^ - w) has values which differ only as to the factor - 1, so 

 that the logarithms differ only by the constant iir ; in fact, for -^ < i^„, the 

 imaginary part of ir"' log sin [tv\~^ (w„ - 'w)] is 



iw'^ tan-i[tanh (ttA"' (-fo - •»!-•)) cot \tt\-' (<j,„- <p)}], 



which increases by i as cjt increases by X, sfr being kept constant. 



From this it appears that the algebraic sum of the integrals along the 

 sides (ji = a, ijt = a+ X, is 

 . « + it 

 f\w)dw, or i j /(a + iO -/(«)!• (42) 



a 



In considering, lastly, the integral yt along tlie line ifr = t, it is to be 

 remembered that the subject of integration has a discontinuity owing to the 

 fact that, at the beginning of the cut to w„, 



ir~ ' log sin {ttA"' (Wj - w) } 



is less by 2i for (j, just less than (ji^ than for ijt just greater than <pa. If con- 

 tinuity be restored to the logarithm, for the purpose of this particular integ- 

 ration, by continuing its analytical form from that for <p > f^, the integral 

 along this side of the rectangle is 



« + t^ a + it 



TT-' log sin I ttX-' (iv„ - w)}f'{w) div - 2i j f'{w) dw. (43) 



a + K + it *'<()„ + •< 



All singularities having been kept outside the contour, the complete con- 

 tour integral must be zero. On combination of formulae (38), (40), (41), (42), 



IJ.I.A. PKOC, VOL. XXKIII , SHOT. A. [8] 



