412 



Meeting of February 28th, 1853 (see the Proceedings of that 

 date). And thus we might prove in a new way the indefinite 

 tendency of the points Q, R', . . on the fixed chord, and there- 

 fore also the corresponding tendency of the points Q, R, . . 

 in the plane, to coincidence with the fixed point B (that point' 

 being still supposed to be real). 



9. By placing Z alternately at G and at H, it may be 

 shewn, in like manner, that the alternate lines PQ, MS', 

 T U, . . all pass through one fixed point, namely, the point 

 where GO' intersects (M) again, after meeting it in the sum- 

 mit G; the other alternate lines Q_R', ST', . . all pass through 

 another fixed point, namely, the second intersection of HO' 

 with (N) ; again, R QJ, TS', . . pass through the analogous 

 intersection of G O" with (M) ; and QP, SR', . . through 

 that of HO" with (N). The opposite summits E, G, H' 

 might be employed in the same way to furnish other theorems, 

 which would not, however, be essentially different from these. 



1 0. ( Second Case : LI > LD). — When the given line LI 

 is greater than LD, or than the radius of the circle (L), that 

 circle is no longer met by the line 0"IO' in any real points, 

 A, B ; but it is obvious, from the known principles of modern 

 geometry, that this latter line is still the common radical axis 

 of three circles (L)(M)(N), whereof the two latter have still 

 theu: centres M and N harmonic conjugates with respect to 

 the given circle (JL), and are still the loci of the two systems 

 of alternate points, P, R, T, . . and Q, S, U, . . . namely, 

 the assumed point and those derived from it by the rule stated 

 in Art. 1, taken alternately : because that rule did not involve 

 any reference to the points of intersection A and B. These 

 circular loci will still have real summits G, H, which will still 

 serve to determine real points, P, Q, R', ... upon the radi- 

 cal axis, by the alternate lines GP, HQ, GR, . . . ; and the 

 same relations of homography and involution will still hold 

 good, conducting to the same theorems of real intersections of 

 lines as before, although the points A and B on the circle (L) 



