
502 Journal of the Asiatic Society of Bengal. [November, 1908. 
which contains ie Resor terms and the necessary number of 
arbitrary constant 
Therefore, the cubic ne Mon aint same contact at any point 
(a, y) of a curve, is of the 
a(X—a)8+ Bi een ai 8(¥-y)(X-2)? 
+A{( Y-—y)da-—(X—2)dy}? 
—2y{(¥—y)de—(X—a)dy}{(¥—y)@a—(X—2) dy} 
+{(Y—y)d’a—(X—2) dy}? -2Q{(Y—y)de—(X—«)dy}=0 (63) 
In general, the equation of a curve of the nt degree, which has 
three-pointic contact with a given curve — the origin will have 
the portion below third degree, of the f 
M{ Yd — Xdy}? —2p{ Yda - oe {Yd’a— Xd?y} 
+ {Yd’2— Xd’y}*? —2Q{ Yde — Xdy} =0 (64) 
It is easy to deduce from the general equation of a conic 
19, 
of three or four-pointic pees — of a four or five-pointic con- 
tact, and the method is a 
For example, the aaa sees of a parabola of three- 
pointic contact is (58) 
((Y¥—y)(d*x — pda) —(X—2) (d*y —pdy)}? 
=2Q{(¥-y)de—(X—a)dy) 
If this parabola meet the curve again at an adjacent point 
(X, Y), then 
1 1 
ails matsneeyhe 3. 
X=a+de+y 34 % +7534 2+ &e. 
eas : (65) 
Y=yt dytigty+i-5 

ay + &e. 
Substituting (65) in (58) and remembering that » is an infini- 
tesimal of first order, we have 
C+ oe 
Again, to determine A, so that we may get the conic of five-pointic 
contact, from the system of four-pointic (56), 
((¥—y)(8Qd?a — Rdx) — (X —2) (3Qd*y — Rdy) }? 
+A{(Y—y)de—(X—2)dy}?=18Q3{(Y¥—y)de—(X-2)dy} 
_ Substitute (65) in (56), and remembering that A is an infini- 
tesimal of order eight, we have 

