188 
ON ORTHOMORPHOSIS. 
[920 
where a, /3 are interior points, the most simple case is when a = a, ¡3 = — a, positive 
and < 1. Here 
z 2 — a‘ 
and we have 
2 , 2 = (# 2 + y 2 ) 2 — 2g 2 (æ 2 — y 2 ) + a 4 
1 ^ 1 — 2a 2 (x 2 — y 2 ) + a 4 (x 2 + y 2 ) 2 ’ 
or say 
which last form shows that to the circumference ¿r x 2 + y? — 1 = 0, there corresponds the 
circumference x 2 + y 2 — 1 = 0 (and besides this, only the imaginary curve x 2 + y 2 + 1 = 0). 
Writing x 2 + y 2 = c 2 , we have the contour 
(c 2 a 4 — 1) (x 2 + y 2 ) 2 — 2a 2 (c 2 — 1) {x 2 — y 2 ) + c 2 - a 4 = 0, 
a bicircular quartic. For c = 1, we have as already mentioned the circle x 2 + y 2 —1 = 0; 
for c less than 1, a curve lying within this circle, diminishing with c, and after a 
time acquiring on each side of the axis of x an indentation or assuming an hour 
glass form; for the value c 2 = a 4 , the equation becomes 
(a 4 +1) (x 2 + y 2 ) 2 — 2a 2 (x 2 — y 2 ) = 0, 
and the curve is a figure of eight, the two loops enclosing the points y = 0, x = a 
and x = — a respectively; and as c 2 diminishes to zero, the curve consists of two 
detached ovals lying within the two loops of the figure of eight, and ultimately 
reducing themselves to the two points respectively. There is no difficulty in finding 
the equation of the curves corresponding to a radius x 1 — 6y 1 = 0, but the configuration 
of these curves is at once seen from that of the former system. We may in the 
present case say that the circle is squarewise contracted to the figure of eight; and 
then further that each loop of the figure of eight is squarewise contracted to a 
point; but we do not have a squarewise contraction of the circle to a point. 
30. A closed curve or contour may be squarewise contracted not into a point 
but into a finite line: we see this in the case of a system of confocal ellipses, 
which gives the contraction of an ellipse into the thin ellipse which is the finite 
line joining the two foci. There is a like contraction of the circle; this is, in fact, 
given by the formula due to Schwarz, “ Ueber einige Abbildungsaufgaben,” Grelle, 
t. lxx. (1869), pp. 105—120 (see p. 115) for the orthomorphosis of the ellipse into 
a circle: this is 
where, if a 2 —b 2 = 1 and a = cos —^ , or, what is the same thing, ib = sin , then 
0(j '2/ 
the ellipse 1 J — 1 + e —™'~ 2 1 - ,2 =0. The contours 
and trajectories for the ellipse are the confocal ellipses and hyperbolas respectively;