921] 
and thence 
hence 
ON SOME PROBLEMS OF ORTHOMORPHOSIS. 
Xj 2 + Y 2 - = r ~ (if %i+ F a 2 be put = r 2 ) ; 
P 2 (1 — k 2 Q 2 r 2 ) = r 2 — Q 2 , Q 2 (1 — k 2 P 2 r' 2 ) = r 2 — P 2 . 
199 
Now considering in figure 1 any line in the rectangle parallel to the axis of X, 
that is, taking F constant and therefore also Q constant, and proceeding to eliminate 
P, we have 
pa _ r " Q" i _ p 2 _ f + Q 2 ~ (i + k' 2 Q 2 ) p 
l-k 2 Q 2 r 2 ’ l-k 2 Q 2 r 2 
1 _ ¿up. = l+^(?-£ 2 (l-H3 2 )r 2 1 t ? 2 ^ _ 1 - k-Q 4 
and consequently 
1 — k 2 Q 2 r 2 
k 2 Q 2 
, 1 + k 2 P 2 Q 2 = 
1 — k 2 Q 2 r 2 ’ 
r. 
QVl + Q 2 -(l + ¿ 2 Q 2 ) r 2 Vi + ^ 2 Q 2 -P 2 (1 +Q 2 ) r 2 
1 — k 2 Q 4 
giving X a , Fi each of them in terms of Q 2 and r 2 , = X 1 2 +F 1 2 . The former of these 
equations, replacing therein r 2 by its value, gives easily 
where 
{X 2 + Y 2 ) 2 - 2AX 2 - 2BY, 2 + P = 0, 
Ar 
O 1 - 1 + Q 2 | 1 1 + k~Q 2 QD A; i 1 ■ 
1 + ¿ 2 Q 2 + A; 2 Ï + Q 2 ’ + & 2 Q 2 ’ 
viz. we have thus the equation of the curve, a bicircular quartic which in figure 2 
corresponds to the line parallel to the axis of X x in figure 1. 
In particular, for the line LM of figure 1 we have F = \K' and thence 
1 
k 
iQ — sn \iK' = , that is, Q = —., and thence A=B — \-, the equation of the bicircular 
wk \/k b 
quartic is 
viz. this is the circle Xj 2 + F x 2 — ^ = 0 twice repeated, and we have thus this circle, 
or rather the half-circumference LFM of figure 2, corresponding to the line LAI of 
figure 1. More simply, 
Xj -tXFj = sn (X + liK') = 
sn X+ j- en X dn X , - — 
Vk V* _ (1 + k)P + iYl — P 2 .1 — k 2 P 2 
\+k 2 . y sn 2 X 
K 
\p (1 + kp 2 )