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 

 ofn points PQR . . , arranged half on one locus, and half 

 upon the other. For example, if LI = 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 six points, 

 PQRSTU, three (PR T) on one locus, and three (QSU) 

 on the other. But if n be odd (for instance, if EX be the side 

 of a regular pentagon), then the result of n 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 (2V) : and the period therefore 

 now consists of 2n points in the plane, whereof n are on the 

 circle (M), and the n others on the alternate circle (N). 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). — In the intermediate case, 

 where the given line LI is equal to LD, the radical axis be- 

 comes a common tangent at E 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 (L) ; and therefore the points 

 Q, R', S', . . . and in like manner R', Q, P', . . . and conse- 

 quently also the points Q, R, S, . . . and R, Q, P, . . . (the 

 lines GPP',HQQ', &c. being now obtained by lines drawn 

 from the summits G and H remote from the common summit 

 E), must all indefinitely tend to that fixed position E as a 

 limit. As regards the law of this tendency, it may be ex- 

 pressed by either of the formulae 



Q!0. 0"P' = EO*; EP. EQ=E O'. Q'P' ; (15) 

 or more clearly by the following, 



E^~E } p = WO^ COnSt - < 16 > 



