500 Journal of the Asiatic Society of Bengal. [November, 1908. 
. The conic of four-pointic contact, at any point (a, y) of 
a tgs curve, has the first, second and third differentials of # and 
e same as with the given curve, but the fourth and higher 
differentials arbitrary, and, in general, different from those with 
the given curve. Hence if we put, in equation (52), 
8QS —5R?+12QR'=d (55) 
where A is an arbitrary constant, we shall have, as the equation of 
the system of conics, of four-pointic contact, at any point (#, y) of 
a given curve, 
{(Y¥—y)(8Qd°e ~ Rdz) - (X—2)(3Qd’y — Rdy) 
+A{(¥—y) da —(X—2x) dy}? =18Q%{(Y—y)da—(X—a#)dy}? (56) 
Again, if wo oe sere and higher differentials of # and y 
arbitrary, and put 3 ooh sain” where p and v are arbitrary 
constants, we have as the equation of the system of conics of three- 
pointic- contact, at any point (a, y) of a given curve, 
{(Y—y)(@a— pdx) —(X—2) (d’y —udy)} 
+v{(Y¥—y)da -(X—«)dy}?=2Q{(Y-—y)de—-(X—«)dy} (57) 
In particular, the equation of the system of parabolas of 
three-pointic contact is 
{ (Y¥—y)(d®e—pdz) —(X—w)(d%y ~pdy)}2=2Q {(Y¥—-y)dx —(X—«a)dy} (58) 
iy Bae | may be interesting to deduce directly the equation 
of a conic of thre ree-pointic contact, from a special form of the 
eer of a conic passing throngh tives given points. 
et (2, y), (2, FMA (22, y,) be the co-ordinates of any three 
BEN ”, re P,, a 
Fe Ps ee 
M=(Y- —y))(#; —2%,) — (X—%)(y2- Y1) ; (59) 
N=( Y~y)(a,—2) —(X—2)(y,-y) 
te equations of the lines PP,, P,P, and PP,, respectively. 
en 
a Font SON er ~ A m@))(yg—2y,+y) ) 
tt 9) (Foaling) 5 (60) 
+H Ney, sere — (%,—2)(y)— 
Now, the equation of a conic through P, re P, can evidently be 
written in the form 
ALM —pN(M—L)+(M-L)?—(M+L)(M+L—N)=0 

