411 



bisectors of the angles APB, AQB\ and prolong FC, FD, 

 or the external bisectors of ACB, ADB, to meet the same 

 fixed chord prolonged in O' and O". Then the formula (6) 

 will still hold good, even with attention to the signs of the seg- 

 ments, after changing P, Q, C, D, to F, Q', O', O" ; we have 

 therefore the two following equations between anharmonic 

 ratios, 



{ABPv>) = {ABQO), (ABFO") = (ABQJ<x>), (7) 

 which give 



gR^Q^^o^A 



AO' FB O'P* K) 



and consequently, 



Q'O' . 0"F = AO'. O'A = IO' 2 - I A* = const., (9) 



where the constant may be variously transformed: for in- 

 stance we may write, 



Q'O'. 0"F=2FI.LI. (10) 



8. The equations (7) shew that we have the two involu- 

 tions, 



(AB, FO\ Q'oo) and (AB, F oo, Q'O"); (11) 



if then from any point Z, assumed at pleasure on any one of 

 the three circles, Ave draw three successive chords of that circle, 

 ZZ' through F, Z'Z" through QJ, and Z" Z" through O', or 

 else ZZ through QJ, ZZ" through F, &aAZ"Z" through O", 

 the fourth or closing chord Z'"Z will in each case pass through 

 infinity ; or in other words, this closing chord will be parallel 

 to the fixed chord AB. In particular, by placing Z, and there- 

 fore also Z" at F, which will oblige Z" to be at C or at D, we 

 see that the lines FF, CQJ (or FQJ, CR', &c.) must intersect 

 each other (at an angle of 45°) in some point Z of the given 

 circle (L) ; and that the same thing holds for the lines FQJ, 

 DF (or FR\ D QJ, &c), as stated to the Academy at the 



