920]
ON ORTHOMORPHOSIS.
189
[920
a, positive
sponds the
y 2 + 1 = 0).
■ y 2 -1 = 0;
nd after a
an hour-
= 0, x = a
its of two
ultimately
in finding
mfiguration
lay in the
eight; and
acted to a
,o a point
al ellipses,
the finite
is, in fact,
en,” Creile,
ellipse into
iirK'
4 K
then
he contours
espectively ;
and for the circle, they are two sets of bicircular quartics, such that the portions
within the circle have a configuration resembling that of the confocal ellipses and
hyperbolas within the ellipse. Po investigate the formulae, it is convenient to introduce
the function
X + i Y, = sin -1 (x + iy) ;
we then have
or say
x + iy = sin (X + iY),
x = sin X cos i Y, iy = cos X sin i Y,
so that to any given value of F there corresponds the ellipse
y 2
and then
or if
— .j-
cos 2 i Y — sin 2 iY
2 K
= 1,
«i + %i = sn — (X + » F) ;
IT
s = sn
2 KX
IT
2 KX
is x = sn
2 KiY
2KiV
C = cn = V(1 - S 2 ), Cl = cn = V(1 + Si 2 ),
7T 7T V '
, , 2KX , j . , , 2KiY .
d = dn = V(1 — ks 2 ), d 1 = dn = V(1 + hs^),
then
giving
xi =
SC-lCLl
1 + k 2 s 2 s 2 ’
Vi
s^d
1 + k 2 s 2 s 2 ’
2| 2 s 2 + Sl 2
x ' + y ; -
and therefore x^ + yf — ^ = 0 if 1 — ks 2 = 0, that is, if Sj = , or si
since
2 KiY
is 1 = sn
7T
■ e 2 KiY 1 >Kr/
ir = f iK, or
ttK’
4 K ’
hence defining a as above, it appears that the elliptic periphery — + —- = 1 corre
sponds to the circumference x ± 2 + y^ — ^ = 0. By the introduction of X+z'F as above,
the circle and the ellipse are each compared with a rectangle ; the reduction of the
circle to the rectangle, as given by the foregoing equation x 1 + iy 1 = sn — (X+iY),
or what is substantially the same thing by an equation x 1 + iy x = sn (X + iY), is
more fully discussed in my paper, Cayley: “On the Binodal Quartic and the graphical
representation of the Elliptic Functions,” Camb. Phil. Trans., t. xiv. (1889), pp. 484—
494, [891], and in a paper “On some problems of orthomorphosis,” Crelle, t. evil. (1891),
pp. 262—277, [921].
31. The whole theory, and in particular Riemann’s theorem before referred to,
