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' . . , nor of R'Q'P' . . , nor con- 

 sequently of the points QR S . . , nor of R QP . . , to any 

 fixed position. 



1 1 . There may however be, in this second case, when A 

 and B are imaginary, " P" q' i p' <y 



a constant circulation 

 in a period, among 

 the derived points in 

 the plane, or on the 

 axis. For we have 

 now, in the formula 

 (9) (compare fig. 2). 



-IA* = IL*-LA* = IL-- L X* = IX 2 = IW\ ( 1 2) 



if IX be a tangent (now real) from I 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'Q . OP' = Q'O'. O'P = 10* + IW~ = O' W\ (13) 



if P" be so taken on the radical axis that I shall bisect P'P" 

 hence O'WQJ = (?FP"Q'=) IPW; subtracting therefore 

 O'WP' from each, and observing that the triangles WO' I, 

 EFX are equiangular, we obtain the formula, 







E\^/ 





\. X/4C 





/zre 

 Wf 



5£x ' / 



uk. 



\ L 





yfc 



P'WQJ = IO'JV= 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 EX 

 be a side or a diagonal of a regular polygon with n sides in- 

 scribed in (L), n derivations of QJ from P' will answer to one 

 or more complete revolutions of the line JVP, and mil conduct 



