Lees, Electrical Rcsistmice of a Quadrilateral. 



e.g., a square A'B'C'D' the angular points A\ B\ C, D', 

 of which correspond to the points A, B, C, D of the 

 quadrilateral. 



That this is possible may be seen from the fact that 

 the contour AC DA of the triangle J.i^C' in the^' plane, may 

 be transformed into the axis 

 of real quantities in a third 

 plane, the iv plane, the three 

 points ^4, C, D, becoming any 

 three assigned points on that ^^ 

 axis, and the area within the 

 triangle the upper half of the 

 ^v plane.* Similarly the contour 

 A' CD' A' of the triangle in the 

 z' plane may be transformed 

 into the same axis, the points 

 A'C D' becoming the same three points, and the interior 

 the upper half of the %v plane. 



Hence the triangle ACD of the z plane may be 

 transformed into the triangle A' C B' of the z plane.f By 



• Christofifel, .<^««. di Mat. I., p. 95 {1S67). Schwarz, Ges. IVerke, II., 

 p. 65. Forsyth, Theory of finictioiis, p. 541. Love, American Journal of 

 Mathematics, XL, ]5. 164 (1SS9). 



t If the internal angles of the triangle ADC are olt:, ."itt, 77r respectively, 



and those of .-/'// C' , -, -, the transforming equation is obtained by 

 424 ^ ^ ^ 



eliminating w from 



J {-u-aV 



dxv 



Y-''{io~bY-\-M-cY-i 



and 



-. = I ( / 



{w — ay[w-bf{w—c)^ 



Love, i.e., has worked out the integrals for : — 



I ^ I I 



: , y=- 



3 3 



I 



4 



5 = 



I 



4 

 I 



they can be expressed by elliptic functions. In the general case they can be 

 expressed, with certain limitation as to paths of integration, by Abelian 

 functions. Forsyth, T/ieory of Functions, ^. ^^t,. 



