38 ■ Proceedings of t lie Royal Irish Academy. 



Similar use might be made of a periodic curve-factor of zero angular period, 

 the origin in the plane of z being outside the boundary curve. 



The relation between the boundaries corresponding to dz = G (w) dw and 

 to z = G {w) is analogous to the relation between an orbit and its hodo- 

 graph. 



The determination of the form of the boundary is usually easier when the 

 transformation is of the latter type, as miglit be exemplified by taking G to be 

 G'l"ss or G'l''^^- But the advantage, at the present stage of the discussion, 

 of transformations of the type z = G{vj) is that they give representations 

 possessing not only one characteristic but all the characteristics required by 

 the specification of article 1. 



4. Cmulition for the periodicity of z. — In tlie previous article it has been 

 seen that the conditions which must be satisfied by (J include all the con- 

 ditions which must be satisfied by/. The converse theorem, however, is not 

 true, and the diflt'erence is important. 



One characteristic feature of the problem under consideration has been 

 formulated early in article 2. Another characteristic feature is that the 

 boundary (corresponding to i/. = 0) in the z plane is a closed curve, and that 

 the curves which correspond to positive constant values of ^ are also closed 

 curves. In other words z is periodic in w with linear period A. 



If the differential relation dz = (fd%v lead, on integration, to the relation 

 z = F(w), so that F = I Gdw, it is necessary that both C and F be periodic. 

 But the mere periodicity of 6' is not a guarantee of the periodicity of F; for 

 if a constant g (possibly complex) be added to 6, the periodicity of G is not 

 impaired, while a non-periodic part gw is added to F. Thus, in the absence of 

 precaution to the contrary, tliere is always a chance of a periodic G leading 

 to a non-periodic F. In the geometrical interpretation this would mean that 

 the curve in the z plane corresponding to i// = 0, instead of being a single 

 closed loop, would be an infinitely extended periodic curve, necessarily with 

 nodes and loops when the angular period is lir, of the general character, for 

 example, of a nodal trochoid. It may obviously be said of such a curve that 

 dz is periodic, but z is not. 



If be the mean value of G calculated for a fixed value of ■^, zero or 

 positive, and for a range of values of ^ of extent A, then for the corresponding 

 range \Gdw = \(J, and the value of z does not repeat itself unless 6'= 0. 

 Hence there must be added to the conditions which G has to satisfy the 

 requirement that its mean value as here defined must be zero. 



It can be seen that, provided ^^is periodic, the mean value which has been 

 defined is independent of the particular positive constant value assigned to ■,//. 

 For, as G has by hypothesis no singularities in the half-plane of i/- positive, 



