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’, &’, . . 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’, RS’, 
TU, ..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 QR’, ST’, . . all pass through 
another fixed point, namely, the second intersection of HO’ 
with (NV); again, RQ’, TS’, .. pass through the analogous 
intersection of GO” with (M); and QP’, SR’, . . through 
that of HO” with (NV). The opposite summits Z, G’, H’ 
might be employed in the same way to furnish other theorems, 
which would not, however, be essentially different from these. 
10. (Second Case: LI > LD)—When the given line LI 
is greater than LD, or than the radius of the circle (Z.), that 
cirele is no longer met by the line O’JO' in any real points, 
A, B; but it is obvious, from the known principles of modern 
geometry, that this latter line is s¢i// the common radical axis 
of three circles (Z)(M)(N), whereof the two latter have still 
their centres M and N harmonic conjugates with respect to 
the given circle (Z), 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 4 and B. These 
circular loci will still have real summits G, H, which will still 
serve to determine real points, P’, Q, Rf’, .. . upon the radi- 
cal axis, by the alternate limes 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 4 and B on the cirele (L) 
Eee 
