414 
from P’ to P’ again, and therefore also from P to P, if the 
number n be even: in this case, then, there will be a period 
of n points PQR.., arranged half on one locus, and half 
upon the other. For example, if LJ = FE = 2LD, the chord 
EX will be the side of an inscribed hexagon ; and wherever 
P may be assumed, we shall have a period of siz points, 
PQRSTU, three (PRT) on one locus, and three (QSU) 
on the other. Butif 2 be odd (for instance, if ZX be the side 
of a regular pentagon), then the result of derivations gives 
indeed the initial position P’ on the axis, but this position now 
answers in the plane not to the first assumed point P on (M), 
but to a certain other point on (NV): and the period therefore 
now consists of 2” points in the plane, whereof” are on the 
circle (M), and the » others on the alternate circle (NV). An 
outline of these results respecting periods of points was lately 
submitted to the Academy, in the communication of last Fe- 
bruary. 
12. (Third Case: LI = LD).—%n the intermediate case, 
where the given line LJ is equal to LD, the radical axis be- 
comes a common tangent at EZ to the three circles, the centres 
M, N being harmonic conjugates as before; and because all 
former theorems respecting intersections of lines hold good, 
the lines FP’, CQ still cross on (Z); and therefore the points 
Q, FR’, S, ... and in like manner R’, Q, P’, ... and conse- 
quently also the points Q, Rk, S,... and &, Q, P,... (the 
lines GPP’, HQQ, &c. being now obtained by lines drawn 
from the summits Gand Hremote from the common summit 
E), must all indefinitely tend to that fixed position Z as a 
limit. As regards the law of this tendency, it may be ex- 
pressed by either of the formulz 
Q'0. O'P'= E0®; EP.EQ=EO.Q'P; (15) 
or more clearly by the following, 
E , EP’ EO’ . (16) 
