4^' Proceedings of the Royal Irish Academy. 



exponential term of highest order. The exponential order at infinity is closely 

 related to the angular period. 



There being no definite infinities or zeroes of ^in the relevant region, the 

 integral jdd'/fr' taken round any contour is zero. Let the contour be the 

 rectangle formed by the lines •;/- = 0, \p = t, (f) = (p^, <!> = <p^-^ X; since 6' is 

 periodic, the subject of integration is equal at corresponding points of the lines 

 <t> ^ ft, and <j) - <p^ + \, and so the integrals along tliese sides of the rectangle 

 cancel one another. The integral along the length X of the line ;/- = equals 

 - i times the corresponding angular range, and the integral along the length X 

 of the line xp = t equals + i times the angular range for this line, that is the 

 angle between the tangents at the extremities of the corresponding curve in 

 the z plane. Thus it appears that the angular range is the same for all lines 

 of length A parallel to the line -ip = 0, being in fact equal to the angular 

 period of 6^ 



If t be made indefinitely great, the limit value of the integral depends on 

 the term of highest exponential order in the formula 11. If 6' be put equal 

 to A ex-p(-iA-w), then cU'lG = - iMw, and the integral from ^^ + i^i to 

 ^u + X + it is - iNX. So if (^'have its exponential order at infinity i\''and 

 its angular period Q, iA''A = i2. When Q = 27r, N = 2irj\, and therefore 

 the 11 of formula (11) must be unity. 



6. A more compreJiensive forrmda for periodic curve-factors. — As the number 

 of types of periodic curve-factor as yet obtained is small, it is desirable to seek 

 some wider formula which may be used for the extension 

 of the category of known types. Consider a semi-infinite 

 strip of width A in the w plane, say the strip between the 

 lines ^ = - 5 A, tp = \X, on the positive side of the axis 

 of (/,. 



If /'is (i) periodic of linear period A, (ii) free from 

 definite zeroes and infinities in the half-plane of vj, then 

 it is equally general to describe ^' as (i) periodic of linear 



. ar=-j\ Ci/^iX 



period A, (li) free from definite zeroes and infinities in ihe 



FiGLKE 1. 



strip. 



The strip in the 16- plane can be represented conformally upon the half- 

 plane of a new variable by the transformation 



e = csm(nu-IX), (14) 



where c is a real constant. Now ^, having no definite zeroes or infinities in 

 the strip in the w plane, must, when expressed as a function of 6, be free 

 from definite zeroes and infinities in the positive half-plaue of 9. And as 6' 

 is a curve-factor whose range of curviliuearity covers A on the real axis in the 



