54 Proceedings of the Royal Irish Academy. 



exp (4s-2!cX), the function F {tif) is zero. Tlius formula (52) gives 



J ^ C ^ log sin (7rA"'2f - I ?/3) + log sin (irA'-w - J i/3 - Jir) 



- log{sin(2-X-'jc - i^)/sin (2n\-'w + i/3)} 



= C" + log sin ilnk-ho + t/3). 



The corresponding form of (^^ or exp J is proportional to sin (2jrX''jt^ + i^), 

 which is (sx- 



(iii) Let f\a) = cosj2;!3-A"'(a + k-)j, where n is an integer and k a 

 constant, so that /(tc) = (A/2n7r)sin |2nH-A"'(ic + k)}. Here it is to be 

 observed that the mean value oif{a) over a range A is not 2jrA-', but is zero. 

 This renders formula (48) iuapplicable ; in fact, -y, = - 2jrzA"'SP, and the 

 coefficient of ic in formula (52), must be correspondingly modified. In this 

 instance /'{vi) has no infinities at definite points in the strip. But the 

 exponential expression for f (w) contains one term which becomes infinite 

 for ./,->+ X , namely \ exp ! - 2nin\'^{w + k)\, and so 



F{w) = - {l/'lun) exp ; - 27ii;rA"' {ic + k)}. 



Thus formrda (52) assumes, in this instance, the form 



./= - 2i(A/2jjs-)sin|2njrA-'(w + K)! -(A/2n7r)exp \-'27ii7r\-'(w + K)] +C" 



= C - (A/2jia-) exp !2MijrA-'(ff + k)|. 



and the corresponding form of exp ,/ is 



6™ = exp [- (A/27i!r) exp ( 2ni!rA-' (w + k) } ]. (53) 



This is not a periodic curve-factor of the kind which has been aimed at, since 

 it is based upon a form of /'(«>) whose mean value is zero, and tends to a 

 definite limit for d, —^ ~ x instead of becoming infinite of exponential 

 order ?5r;A. It is an inflexional periodic curve-factor whose angular period 

 is zero. 



/mi is, nevertheless, useful for the building up of periodic curve-factors of 

 angular period 2a- ; for, if /'[aj be taken of the form 



27rA-' + c cos {2n7rA-' (a + «)!, 



the corresponding curve-factor is 



^71 = Cs7 GlO, (54) 



and it is clear that (J-.i, or any other product of Gs; with powers of different 

 particular cases of (:-„, is a periodic curve-factor whose angular period is 2n-. 



On this may be founded a general formula. For /'(a) can be expressed 

 as a Fourier series, and to each term of this series there corresponds a cur\-e- 



