413 
have now become imaginary. For example, the lines FP’, 
CQ, or the lines FQ’, DP’, still cross at an angle of 45° in 
some point Z’ on that given circle (Z): but because the radical 
axis-is now beyond that circle, there is now no tendency to any 
convergence of the points Q'R'S’.. , norof R’Q’P’ .., nor con- 
sequently of the points QRS.., nor of RQP .., to any 
Jixed position. 
11. There may however be, in this second case, when 4 
and B are imaginary, 0” Pr’ @rI P By 
a constant circulation VIMY 
in @ period, among NIN 
the derived points in EX aN 
the plane, or on the VAX 
axis. For we have 
now, in the formula 
(9) (compare fig. 2). 7 
— [42 = IL? - LA? = IL? - LX? = 1X*=IW, (12) - 
if IX be a tangent (now real) from J to (L), and if W be one 
of the two fixed points in which the common orthogonals to 
the three circles (now really) intersect each other: thus (9) 
becomes, in the present case, 
O'?Y.OP’=Q00'.0°P = 10°+ IW? =O'W, (18) 
if P” be so taken on the radical axis that J shall bisect P’P” 
hence O'WQ' = (WP’Q' =) IP’'W; subtracting therefore 
O'WP’ from each, and observing that the triangles WO’, 
EFX are equiangular, we obtain the formula, 
PWQ =IOW = EFX = const. (14) 
If then the constant angle thus subtended by P’Q’ at W be 
commensurable with a right angle, or in other words if HX 
be a side or a diagonal of a regular polygon with n sides in- 
scribed in (ZL), derivations of Q’ from P’ will answer to one 
or more complete revolutions of the line WP’, and will conduct 
