ADDRESS. 15 
other, and consequently the motion of the one determines the motion of the 
other: for instance, the two points may be the tracing-point and the 
pencil of a pentagraph. You may with the first point draw any figure 
you please, there will be a corresponding figure drawn by the second 
point: for a good pentagraph, a copy on a different scale (it may be) ; 
for a badly-adjusted pentagraph, a distorted copy: but the one figure 
will always be a sort of copy of the first, so that to each point of the one 
figure there will correspond a point of the other figure. 
In the case above referred to, where one point represents the value 
zw + ty of the imaginary variable and the other the value X + 7Y of some 
function ¢(z + vy) of that variable, there is a remarkable relation between 
the two figures: this is the relation of orthomorphic projection, the same 
which presents itself between a portion of the earth’s surface, and the 
representation thereof by a map on the stereographic projection or on 
Mercator’s projection—viz. any indefinitely small area of the one figure is 
represented in the other figure by an indefinitely small area of the same 
shape. There will possibly be for different parts of the figure great 
variations of scale, but the shape will be unaltered; if for the one area the 
boundary is a circle, then for the other area the boundary will be a 
circle; if for one it is an equilateral triangle, then for the other it will be 
an equilateral triangle. 
I have for simplicity assumed that to each point of either figure there 
corresponds one, and only one, point of the other figure; but the general 
ease is that to each point of either figure there corresponds a determinate 
number of points in the other figure ; and we have thence arising new and 
very complicated relations which I must just refer to. Suppose that to 
each point of the first figure there correspond in the second figure two 
points: say one of them is a red point, the other a blue point; so that, 
speaking roughly, the second figure consists of two copies of the first 
figure, a red copy and a blue copy, the one superimposed on the other. 
But the difficulty is that the two copies cannot be kept distinct from each 
other. If we consider in the first.figure a closed curve of any kind—say, 
for shortness, an oval—this will be in the second figure represented in 
Some cases by a red oval and a blue oval, but in other cases by an oval 
half red and half blue; or, what comes to the same thing, if in the first 
figure we consider a point which moves continuously in any manner, at 
last returning to its original position, and attempt to follow the corre. 
sponding points in the second figure, then it may very well happen that, 
for the corresponding point of either colour, there will be abrupt changes 
of position, or say jumps, from one position to another; so that, to 
obtain in the second figure a continuous path, we must at intervals allow 
the point to change from red to blue, or from blue to red. There are in 
the first figure certain critical points called branch-points ( Verzweigungs- 
pinkie), and a system of lines connecting these, by means of which the 
colours in the second figure are determined; but it is not possible for me 
to go further into the theory at present. The notion of colour has of 
