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921] 
ON SOME PROBLEMS OF ORTHOMORPHOSIS. 
197 
10. It is convenient to collect here the several equations relating to the 
orthomorphosis of the square. We have 
X 1 + xY 1 = sn (Z + iY), k — 3-2V2; 
x x + iy Y = snl {x + iy); 
1 -i(X 2 + iY 2 ) 
'Jk{X x + »70=- 
Z 2 + i F 2 — i 
\'k (X, + *T.) = (» +O + *#■)+ 8 
0 - l)(^i + fy 1 ) + V2 
Z 2 + iY 2 = + »y0> 
which are the equations connecting together the coordinates of the five figures. 
11. I examine more in detail the above-mentioned transformation 
Y . oY — 1 + ^ + . 
As+ ^ 2- X 1+ »Ti + » ’ 
see the foregoing figures 1 (IF) and 2 (Z x F), in which we now regard the two 
circles as having each of them the radius unity; changing the sign of i, the equation 
gives 
Y jy _ 1 — 1 (^i — ^i) 
and we hence find 
Zr+ Fd — 2 Fj + 1 
Z 2 + F 2 = 
Zj 2 + Fj 2 + 2 F x + 1 ’ 
consequently if Z } 2 + F, 2 + 1 = 0, then also Z 2 2 + F, 2 + 1 = 0, or the transformation 
changes the first of these imaginary circles into the second of them: or say it 
changes the concentric orthotomic of the circle X x 2 + Fj 2 — 1 = 0 into the concentric 
orthotomic of the circle Z., 2 + F, 2 — 1 = 0. 
We have moreover 
Zo = 
2Z, 
Fo = 
X 2 + Yj 2 -1 
Z 1 2 +F 1 2 +2F 1 +1’ 2 Z 1 2 +7 1 t + 27 1 + l’ 
values which give the foregoing expression for Z 2 2 + F 2 2 . But we further obtain 
- (Zd + Fj 2 — 1 + 2/xFj) 
Z , 2 + F, 2 - 1 - - 7, = - 
2+ 2 2 X 1 2 +Y 2 + 2Y 1 + l ’ 
and it thus appears that the circumferences 
Zi 2 + Y 2 -1 + 2/xFj = 0, Z., 2 + Y 2 - 1 - - Yo = 0,