26 
[711 
711. 
ON A DIAGRAM CONNECTED WITH THE TRANSFORMATION OF 
ELLIPTIC FUNCTIONS. 
[From the Report of the British Association for the Advancement of Science, (1881), p. 534.] 
The diagram relates to a known theorem, and is constructed as follows. Consider 
the infinite half-plane y = +; draw in it, centre the origin and radius unity, a 
semicircle; and draw the infinite half-lines x — — -J-, and x — | ; then we have a 
region included between the lines, but exterior to the semicircle. The region in 
question may be regarded as a curvilinear triangle, with the angles 60°, 60°, and 0°. 
The region may be moved parallel to itself in the direction of the axis of x, through 
the distance 1; say this is a “ displacement ”; or we may take the “ image ” of the 
region in regard to the semicircle. Performing any number of times, and in any 
order, these two operations of making the displacement and of taking the image, we 
obtain a new region, which is always a curvilinear triangle (bounded by circular 
arcs) and having the angles 60°, 60°, 0°; and the theorem is that the whole series 
of the new regions thus obtained completely covers, without interstices or over 
lapping, the infinite half-plane. The number of regions is infinite, and the size of 
the successive regions diminishes very rapidly. The diagram was a coloured one, 
exhibiting the regions obtained by a few of the successive operations. 
The analytical theorem is that the whole series of transformations, co into 01(0 ^ 
J 7 ft) + o ’ 
where a, ¡3, 7, 8 are integers such that aS — /3y=l, can be obtained by combination 
of the transformations co into w+l and co into . 
CO