190 
ON ORTHOMORPHOSIS. 
[920 
are intimately connected with Cauchy’s theorem, “If a function f{z) is holomorphic 
over a simply connected plane area, and if t denote any point within the area, then 
where z denotes x + iy, and the integral is taken in the positive sense along the 
boundary of the area.” See Briot and Bouquet, Théorie des fonctions elliptiques (Paris, 
1875), p. 186. 
Here in order to obtain by means of the theorem the value of the function 
f (z) for a given point t(= a + ih) within the area, we require to know the values of 
f{z) for the several points of the boundary: viz. if z refers to a point P on the 
boundary, and if we represent the value f{z) by a point P x in a second figure, then 
these points P 1 form a closed curve or boundary in this second figure, and we require 
to know not only the form of this boundary, but also the several points P ± thereof 
which correspond to the several points P of the first-mentioned boundary, or say we 
require to know the correspondence of the two boundaries : this being known, we 
have by the theorem the value of f(t), that is, the point cti-t-ï&j within the second 
area, which corresponds to the point t = a + ib within the first area. The (1, 1) 
correspondence of the two areas is of course implied in the assertion that f (t) has 
a determinate value, determined by means of the given values of f(z) along the 
boundary.