48 Proceedings of the Royal trish Academy. 



The function /(a) must be such as to conform to certain conditions. It is 



clear that /'(a) must (for real values of «) be periodic in a of period A, and it 



is also clear that 



/(a + X) -/(,.) = 27r. (34) 



The electrostatic interpretation of the (^, w) transformation is in terms of 

 an electrified cylindrical conductor alone in its own field ; and it is known 

 that when there is only one conductor tlie charge is of the same sign at all 

 points of its surface. Hence if ds be an element of arc of the boundary the 

 sign of dalds is everywhere the same, so that d')^/da and d^/ds have every- 

 where the same or everywhere opposite signs. Thus if the curve be every- 

 where convex,/' (a) must be always of one sign, say positive. But if the 

 curve may have concave parts, /'(a) is not so restricted. 



Formula (34) indicates that the mean value of /'(a) must be 'In/X. 



One way of summing up the requirements of the function f'(a) is to say 

 that it is capable of being represented by a Fourier series corresponding to a 

 wave-length A, that the absolute term in the series is 27r/A, and that there are 

 no terms in sin (27ra/A) or cos {'Iira/X). Thus, for example, 



27r/A + c cos (47ra/A) 



is a possible form of /'(a). 



A particular kind of geometrical consideration may be useful in suggesting 



possible forms of /'(a). Consider any closed plane curve (not to be confused 



with the boundary curve in the z plane), whose tangent makes an angle oj with 



a fixed direction in its plane. Let ds be the element of arc ; then it is known 



that 



ds 



- cos CO da) = 0, 



dut 



,— sinoj dw = 0, (35) 



d<s) 



for a range of tu of extent '2w. Let a new variable a be defined by the rela- 

 tion to = 'InajX. The radius of curvature dsldw is a function of (u ; let its 

 form be 



dsjdw = XfiXcoJ-Zw) . (36) 



This defines a function /'(«) which is periodic anil, in virtue of the relations 

 (35), satisfies the conditions (33). In order to satisfy the condition (34) it is 

 only necessary to choose the linear dimensions of the curve so that the peri- 

 meter shall be iw^. 



For example, in the ellipse (a, h) 



ds/dco (X. (a° cos' co + ¥ sin' QJ)"^'^ 

 and therefore 



A |«= cosX27ra/A) + b' sinX2iralX)}~'l' (37) 



is a possible form of /'(a), provided the constant A be suitably adjusted. 



