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411 
bisectors of the angles APB, AQB; and prolong FC, FD, 
or the external bisectors of 4CB, 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 P’, Q’, O', O”; we have 
therefore the two following equations between anharmonic 
ratios, 
(ABP &) = (ABQ’0/), (ABP’0") = (ABQ), (7) 
which give 
O'- PB~ OP” (8) 
and consequently, 
Q’0'. O’P’ = AO’. O’A = IO? - IA? = const., (9) 
where the constant may be variously transformed: for in- 
stance we may write, 
QO’. O'P’ = 2FI. LI. (10) 
8. The equations (7) shew that we have the two involu- 
tions, 
(AB, P'0', Q’) and (AB, P’~, QO"); (11) 
if then from any point Z, assumed at pleasure on any one of 
the three circles, we draw three successive chords of that circle, 
ZZ’ through P’, Z’Z" through Q’, and Z”Z” through O’, or 
else ZZ’ through Q’, Z’Z” through P’, and Z”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. Inparticular, 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 FP’, CQ’ (or FQ’,CR’, &c.) must intersect 
each other (at an angle of 45°) in some point Z’ of the given 
circle (Z); and that the same thing holds for the lines FQ’, 
DP’ (or FR’, DQ’, &e.), as stated to the Academy at the 
