192 
ON SOME PROBLEMS OF ORTHOMORPHOSIS. 
[921 
= K*Jk, where k has the foregoing special value: I notice the numerical values 
\ = 1-311028, & = 3 - 2 V2 = sin 9° 52', k = 1-582548. The relation between the lemnis 
cate function snl, and the sn with the foregoing value of k, is easily shown to be 
V&sn = where = 
(t-l)snltt + V2 2 \/k 
and it may be added that we have 
V2 
cn U = 
(i — 1) snl u + V2 
A /-^ V(1 + snl w) (1 — i snl u), 
dn U = 
V2 
(i — 1) snl u + V~2 
Vi -1- A; V(1 — snl u) (1 + i snl u). 
2. The Schwarzian orthomorphosis of the rectangle into the infinite half-plane 
is given (Memoir, p. 113) by the formula X 1 + iY 1 = sn (X + iY), where the modulus 
is real, positive, and less than unity. Here, see the figures 1 (XY) and 2(M 1 F 1 ), 
Fig. l (AT). 
Fig. 2 (A 1 F 1 ). 
the rectangle A.SCD, the sides of which are AB — 2X and BC — K, is transformed 
into the upper infinite half-plane (Ti = +), the four corners of the rectangle corre 
sponding to the points A, B, G, D on the axis of X, where OB(=OA)— 1, OC(= 0D) = j_. 
3. We can, by a properly determined quasi-inversion (as will be explained), 
transform the X v Fj-figure into a new figure see figure 3 (X 2 F 2 ), the infinite AVaxis 
being transformed into the circumference of a circle (the radius of which may be 
taken to be = 1) and the infinite half-plane into the area within the circle. The 
four points A, B, C, D are thus transformed into points on the circle, which if the 
quasi-inversion be a symmetrical one, will be situate, A and B symmetrically, and