
a ee eee 
Vol. IV, No. 4.]. A General Theory of Osculating Conics. 175 
[N.S.] 
Eliminating © 
send the above two equations, we have 
A= 3 qs — 5r8 
(36). 
refore the co-ordinates of the centre of the conic of 
closest contact are 
Sou oe 
(398-578) {(Y¥=y)—p(X-w)}* + 
— (X=2) (pr—848)}*= 
3qs —578 
— 27% 

3qs — 57% 
and the equation of the conic of closest contact is 
{((Y—-y)r 
189° {((Y-y)—p (X—a#)} (38). 
Therefore the conic of ear contact is an ellipse, hyperbola 
or parabola, oleae as 3qs 
ll. It may be mrberontdng a deduce the equation of the conic 
of closest ‘erable directly by the method of general differentials. 
The general equation of a conic through (#, y) and (2, y,) is 
of the form, already given (1), viz., 
A (X—a)(H—a) +0 (Foy)\(L~m) + (Xoey(L 
4 is positive, negative or zero. 
yi) 
p(¥— y(X— #)=0, 
herefore the conic ae any five points (2, y), (%, y)); 
(%,, ey (ag. Yu), (yy Y4), 18 
(X—2)(X—2)) 
(#a—2)(22—2)) 
(23—2)(@3—2)) 
(@y~-2) (2-2) 
or 
(KX—«)(X—«) 
(22 —~2#)(2— 2)) 
(%—@)(#3—z%)) 
(ag—2)(%y—2 }) 
<i 
(Y-—y)(Y¥-y) 
(ya—y(va—m) 
(vg—¥) ¥g~y1) 
(va—¥)(ve— v1) 
(¥—y)(Y—y,) 
(yo—y)(ya— 1) 
(Y¥s—y)'va~Y1) 
(Ye—y)(Ys— 1) 
(X—2)(Y¥~—y;) 
(22—2)(ya— yi) 
(eg—2)(yg—y1) 
(%y—#)(yy— 41) 
‘(X—2)(Y¥-y) 
(w2—2)(y2—Yy)) 
(2g—a) yg—y1) 
(t4—2)(y¥g— 4) 
(Y¥—-y)(X—2)) 
(ya—y)e2— 21) 
(y3—yNeg— 2) 
(yg—y)(%4—#1) 
=@0 
(Y-y) 2-2) — (X-2)(n-y) 
ee eed <0 (39 
(vs—v) (1 —2) —(9—2) 1 = 
(yg- y)(a1- 2) — eaten be 
Now if (2, y)s (2s y1)s (ay Ya)o, (2a, Yas (a Ys) be five con- 
secutive points on a curve, then, as in (3), 
2, =2+ Ldn + Be 
t= 2+ 4dx+ 6d*x + 4d?u + dix 
with corresponding expressions for 4j;-Y2. Ya. Ys: 
Br e+ Bde +3d¥e+ dx 
q (40) 
On making substitutions (40) in (39), we have, after ama? 
fication of the determinant by adding to the third row, the secon 
