
Psi 
Vol. IV, No. 10.] A General Theory of Osculating Conies. 501 
[N.S.] 
where A and » are arbitrary constants, for, it is the same as 
A LM-—p(MN-NL)-(4LM—MN-NL)=0 
which circumscribes L=0, M=0, N=0 
Thus, the Sag equation of a conic, through three given 
points, is of the for 
A{(¥-y)(#, —%) —(X-m)(y,-y)}{(F—-y) (a, - 2) 
—(X-2a)(ys-%)} 
—a{(¥—y) (@,—#) —(X—2)(y,-y)}{(¥ -y1) (a — 2a, +2) 
~(X-2%)(y2—-2y, +y)} 
+{(¥ -y;) (@,- 2a, +2) -(X—a%)(yz—-2y+y)VP 
—{(¥-y,)(@,-#) —(X—%)(yg—y)}{(yo—y) (1 —2) 
— (#2, —-#)(y~y)J=O (61) 
Nowif (2, y), (@, y)s (@z, yg) be consecutive points on a curve then 
a, =e+dz, x,=%,+ dz,=2 + 2det ds ‘ 
nay t dy, y2=y,t dy =y +2ly + dy 
Therefore (61) becomes 
A{(Y—y)dw—(X —«)dy}* —2n{( Y-—y)dx—(X—2)dy}{ (¥-y)d*z 
— (X—z)d*y} 
+ {(Y—y)@a—(X ~ )d'y}*-2Q{(Y—y)da + (X—«)dy}=0 
Or {(¥—y) (de — pdx) — (X—2)(d*y — pdy)}* 
+v{( Y—y)da—(X ~2x)dy}*=2Q((Y—y)dx—(X—2)dy} 
where v=A—yp*. This equation is the same as (57). 
. Again, the general equation of a cubic through three 
given points (2,y), (%,4)), (@Y2g) can evidently be written in the 
en @(X—9)(X—a,)(X=a) + A(X -y)(Y—m)(F—y) 
+y(X—2)(Y—y;)(Y—y,) + 8( ¥—-y)(X—2,)(X—«,) 
+A{(Y¥—y,)(#,-—#) — (X-2,)(y,-y)(Y-y)(@—- 21) 
—(X-2#)(y,;—41)3 
—w{(¥-y)(#,-#) —-(X—2)(y,—y)} 
{(Y—y)(#,— 2a, +2) —(X-a%)(y,-2y,+9)3 
+{(F—y)(a%— 2a, +2) —(X-m)(y.—2y, + y)P 
—{(Y—y;)(#,-—2z) -(X—2%)(y,—y)} 
{(ys—y)(#1- 2) — (@-2#)(y,—y)} =0 (62) 
