relative to the Hexagon inscribed in a Conic Section. 169 
: — C ie le See ee 1 
(AB) atcomnao% (1.) 
Naess ie ae eye.) aes (2.) 
Ge age 
Y B-Y¥o os 
(D E) gr aoe inline a coke (3.) 
y Bir" = 
(F A) ww iB; CAG rad = Wale Wey Pars (4.) 
Subtracting (3.) from (1.) and (4.) from (2.), we obtain the 
equations of G Hand HK. 
(GH)...{2- ae ead Bg bY = Oo(5) 
(HK)... {2 - Bot le + + {Saf by = 0.06.) 
And if XY be the coordinates of H, and X, Y, those of 
a then from (5.) and (6.) we get 
a (4%, + Bx,—a, B) x, aad 2 (4 Y.+B, %o—a By) 2, f 
Y (4 Yet Bxa—% B)y, — XY (44 +8, 4-48) yo 
Now the coordinates of the points C, B, E, F, A, D being 
substituted in the equation of the conic section, 
Prtbeyt+cxr?+dy+ex+f=0 
will give the six subsequent equations, viz. 
Ce +ea+ f= 0. (7) | f+ dB+f= 
Ca*+ea+f=0... (8.) | 6% +4d6,4+f= 1 
yy thay, +er,2+dy,+e7,+ f=0.... ie 
Yo +6 Xo Ya+e%, +dy,+ett+f=o....( 
From (7.), (8.), (9.), (10.) we derive 
e=—c(4+a,); f=caa,; d= — (6+8,), andf= 68, 
BB, ee 
aA) 
Multiply equation (11.) by xv, y5 o a ) by v, ¥,, and sub- 
tract; then we get 
or 
and (ad 
~~. C= 
(a+). 
YoY ter +dy,+ex, tf)=2y; (Yo +e ty +dy,+ex,+ f), 
which by substituting the preceding values of c, d,¢, f is trans- 
formed to 
HY _ em ye +h By, e?—a ot; (8+) —B By (4—4)) a+ w, B A, 
r2Y2 6h, YP +B, wy —e @ (B+) Y2o—B Ai (@ +e) to+e a BB, 
