202 
ON SOME PROBLEMS OF ORTHOMORPHOSIS. 
[921 
that is, 
jo-! 2 + Vi) 2 - {(«1 + ?/i)' 2 - c 2 } + 4,cWyi = 0. 
Writing for a moment + y a 2 = P, x li y 1 = Q, this is 
ji» - i (P + 2Q)| (P + 2Q - c ! ) + = 0, 
an equation which contains the factor P + 2Q; throwing this out, the equation becomes, 
after an easy reduction, 
(P -1)* + (1 - c ! ) jp- 2Q —L (P + 2Q)j = 0, 
that is, 
Oi 2 + yi> - l) 2 + (1 - c 2 ) jc«! - 2/ x ) 2 - * 2 (a?a + 2/i) 2 | = 0, 
the required equation. Transforming through an angle of 45° by writing 
Xj = 
OCj+lh 
V2 ’ 
V i = 
x 2 y 2 
Vl 
(where observe that the axis of # 2 is the line PIT of figure 5), the equation becomes 
(x 2 2 + y 2 - 1) 2 + (1 - G-) {2y.? - 2x/j = 0, 
or writing c = cos 7 and therefore 1 — c 2 = sin 2 7, this equation becomes 
(xf + y 2 2 ) 2 - +* 2 - 2 cos 2 7 . y 2 2 + 1 = 0, 
a curve consisting of two indented ovals situate symmetrically in regard to the axis 
Fy 2 of figure 5. In fact, writing in the equation y 2 = 0, we have for x 2 2 two real 
positive values; but, writing x 2 = 0, we have for y 2 2 two imaginary values. For 7=0, 
the equation becomes 
{xi + y/f - 2 (x 2 + yi) + 1=0, 
that is, we have the circle x 2 2 + y 2 2 —1 = 0 twice repeated. One of the ovals is 
shown in figure 5; the portion of it lying within the circle agrees with Schwarz’s 
figure, p. 113, turning this round through an angle of 45°. 
For the lines x—y = G of figure 4, we have in figure 5 the same system of 
bicircular quartics turned round through an angle of 90°. 
II. 
15. I consider the general problem of the orthomorphosis of a circle into a 
circle: we can, for the transformation of the circumference of the circle x 2 + y 2 — 1=0 
into that of the circle x-? + yy — 1 = 0, find a formula involving an arbitrary function