46 Proceedings of the Roijal Irish Academy. 



Thus the condition for periodic z, that is, for a closed triangle, is 



1 - -)exp;-^^) = 0. (24j 



This is, of course, equivalent to the two conditions 



and these are equivalent to the statement that a triangle can be drawn such 

 that the directions of its sides make the angles 



27ra/X, 27r/3/A, 27r7/X, 

 with a fixed direction, and the lengths of the sides are proportional to 

 1 - {Ah). 1 - {Bh). 1 - (6'/7r). 



All the above argument applies equally to a polygon whose internal angles 

 are A, B, C, D, &.Q.., provided each of the summations be understood to include 

 as many terms as there are comers. 



9. Focal 2}cr iodic cu nx-f ad ors. — The transformation of the previous article, 

 giving the field outside a polygonal boundary, may be interpreted as giving 

 the field round a different boundary got by assigning to \p a. constant positive 

 value. Such new boundary would be smooth. A new \p, chosen to vanish on 

 the new boundary, is introduced by substituting \p + k for ^, or vj + Ik for vj, 

 ill formula (22), it being understood that k is positive. Thus there is obtained 

 what may be called a focal periodic curve-factor, 



C"cT = n [sin |7r(»- + u - a)/A]]'-7, (2G) 



wlierein 



\^d,., a„a 2(l-^)expp^«)=0. P7, 



The corners of the polygon, now outside the relevant region, may be called 

 foci of </(,;, and any number of foci may be introduced. When there are two 

 foci, A = Q, B = Q, and the condition for periodicity of z is 



exp (- 2i7r<iA) + exp (- 2i7r/3/A) = 0, 

 which is equivalent to /3 = a + iX. So the bifocal curve-factor is seen to be 

 practically equivalent to df. 



In t^^, it is understood that k is the same in all the factors of the product, 

 but if, instead, the constant typified by k be different in the different factors, 

 and denoted by a', /3', &c., there results the more general function 



C^^ = n [sin [7r(u- - a + ia')/A 1 ]'-7, (28) 



Bubject to the conditions 



2{.-^J = ^. 2(l-^)e.p{-??(«-..)S = 0. 



(29) 



