Lka'I'Hem — Periodic Conformal Curve-Factors and Corner- Factors. 53 



Consequently 



o + A + i* .n + A4«< 



TT-' log sin { ttX-' («'o - w) j /'(w) c^w = CA + Ai^(w„) - lA" 





-tA-'(w-w„)/(7/_;) 



a + A + i* .0 + A + i< 



+ i\- 



/(w) f^w + CA + A i?' (wj . (50) 



a + it •' a + it 



In the last integral f(w) may be replaced by y„tu without altering the 

 result of the integration, and in the immediately preceding expression 

 it is to be noted that f{a+X + it)=^y„\^f{a + it). Thus the whole 

 expression reduces to 



CA -ifia -i it) + iy„w„ - |iy„A + \F{^v„). (51) 



The substitution of this in formula (44,i leads to 



J = 2t2[Plogsin{7rA-'(w- w',)} - ■n-\''Qcot{TrK'Hvj -to,)] + etc.] 



- 2if(io) + 27ra-i(l - ^iF)ta + \F(w) + C", (52) 



where w has been substituted for v\, and C" represents a constant. 



In this expression it is interesting to note how the infinities under the 

 first S sign cancel the infinities off(iv) at definite points of the vj plane, 

 and how those exponential terms of Ai^^ic) which become infinite for 

 \p ^- + CO cancel the corresponding terms in - 2if{w) ; thus ,_/ has no 

 infinities except an infinity for <//-> + a> corresponding to a term which 

 is linear in to, namely - 2i7rA"'w. 



12. Some particular cases may be considered. 



(i) Let /'(a) be a constant, namely equal to 2;rA"', so that /(w) is 2n-A"^!/j. 

 Then most of the terms in formula (52) vanish, and J differs only by a 

 constant from - 27rtA"'ii) ; so fe'ea or exp J is proportional to exp (- 2iriX-hv). 

 which is (j6T 



(ii) Let 



f\a) - 27rA-» sinh 2/3/ {cosh 2j3 - cos (47rA-ia)l, 

 so that 



/ ('<'] = - 2^ log {si'i (27rX,"'w - i(5)/ain (2irA"'w + t/3) }. 



In a strip of breadth A, /'(w) has two simple poles, namely (if a be chosen 

 within suitable limits) at w = Wi = iX(5/2w and w = Wj = JA + iX(5/27r, and at 

 each of these P = - | i. Thus 1 - 2iP = 0, and the expansion of / (w) for 

 \L great and positive has no term linear in to. As the expansion of /' (w) for 

 \p great and positive has no term of a higher order of magnitude than 



[8*] 



