389-391] 



lines V=V 0i F= 

 for this will be 



Special Problems 343 



0, U = C in the TF-plane. The transformation 



* ...(320), 



where a , a p and a q , a r are the points on the real axis of f which determine 

 the ends of the electrodes. By elimination of f from the integrals of equa- 

 tions (319) and (320) we obtain the transformation required. 



391. The following example of this method is taken from a paper by 

 H. F. Moulton (Proc. Lond. Math. Soc. in. p. 104). 



P Q 



z-plane. 

 FIG. 101. 



TF-plane. 

 FIG. 102. 



In fig. 101, let ABCD be a rectangular plate, the piece PQ of one or more 

 sides being one electrode, and the piece RS of one or more other sides being 

 the other electrode. Let the rectangle PQRS in fig. 102 be its transforma- 

 tion in the TF-plane. In the intermediate f-plane, let the points A, B, C, D 

 transform to % = a, b, c, d respectively, and let the points P, Q, R } S transform 

 to f = p, q, r, s respectively. Then the transformations are 



dW 



-Kf-p) (?-?);-) u;-)] -*. 



If we write 



(b -c)(a- d) _ (q -r)(p- s) 

 (a-c)(b-d)- K ' (p-r)(q-s) * 



2m = V(a - c) (b - d), 2m' = V(p -r)(q- s), 



the integrals are 



.,_ a (b - d) b(a d) sri* mz (mod K) 

 6 d (a d) sn' 2 mz (mod K) 



p(g-s)-q(p-s)su 2 m'W(mod\) ( } 



q - s -(p- s ) sn^'T^mod X) 



T7- nff' 



The sides AB, AD of the first rectangle are the periods , - - of 



sn mz (mod K) ; the sides PQ, RS of the second rectangle are the periods in 



TJT- L iU 



W , say -, , -, , of sn t 

 m m 



