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PRESIDENTIAL ADDRESS TO THE 
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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 x + iy of the 
imaginary variable and the other the value X+iY of some function cj)(x + iy) 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 case 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 corresponding 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 (Verzweigungspwikte), 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 course been introduced only for facility of expression; it 
may be proper to add that in speaking of the two figures I have been following Briot 
and Bouquet rather than Biemann, whose representation of the function of an 
imaginary variable is a different one. 
I have been speaking of an imaginary variable (x + iy), and of a function 
(f> (x + iy) = X + iY of that variable, but the theory may equally well be stated in