921] 
ON SOME PROBLEMS OF ORTHOMORPHOSIS. 
205 
viz. this is a circle, coordinates of centre (2, 0) and radius = V3, cutting the circle 
x 2 + y 2 -1=0 in two real points. Referring to the figures 6 (say) and 7 (x^), and 
observing that x 1 = 0, y 1 — 0, that is, z 1 = 0, gives z — 0, or z — 2, that is, the points 
Fig. 6 (xy). 
Fig. 7 (Xji/j). 
V 
x = 0, y — 0, and x=2, y = 0, we see that to the centre 0 in figure 7, there 
correspond in figure 6 the points 0, M which are the centres of the two circles. 
To any small closed curve, or say any small circle surrounding the point O of 
figure 7, there correspond in figure 6 small closed curves surrounding the points 
0, M respectively; and if in figure 7 the radius of the circle continually increases 
and becomes nearly equal to unity, the closed curves of figure 6 continually increase, 
changing at the same time their forms, and assume the forms shown by the dotted 
lines of figure 6. . It thus appears that, to the whole area of the circle x 2 + y 2 —1 = 0 
of figure 7, there correspond the two lunes ACB and ABD of figure 6; or if we 
attend only to the area included within the circle x 2 + y- — 1 = 0 of this figure, then 
there corresponds not the whole area of this circle, but only the area of the lune 
z (z 2^ 
ACB: and thus that the assumed relation 2 i = _ q—establishes, in fact, an ortho- 
morphosis of the circle x-f + y 2 —1 = 0 into the lune ACB which lies inside the circle 
x^ + y 2 —1 = 0 and outside the circle (x - 2) 2 + y 2 — 3 = 0. It may be added that, to 
the infinite area outside the circle x-f + y 2 -1=0 of figure 7, there correspond in 
figure 6 first the area of the lens AB common to the two circles, and secondly the 
area outside the two circles: we have thus an orthomorphosis of the area outside 
the circle x? + y 2 — 1 = 0 into these two areas respectively. 
A somewhat more elegant example would have been that of the correspondence 
_ z (z — V2) 
Zl ~ 1 — \Î2z 5 
here, corresponding to the circumference x 2 2 + y 2 2 — 1 = 0, we have the two equal 
circumferences x 2 + y 2 — 1=0, and (x - V2) 2 + y 2 — 1 = 0 : and to the whole area of the 
circle x 2 + y^ —1=0, there correspond two equal lunes ACB and ABD.