176 = Jornal of the Asiatic Society of Bengal. [April, 1908. 
row multiplied by (—8), and to the fourth row, the third row 
multiplied by (—4) and the second row multiplied by 6, and by 
ultimately eg all higher orders of infinitesimals 
(X—a)2 (7—¢P 
2(dz)2 2(dy)2 
6dad2x 6dyd2y 
6(d2xz)2 + Bdzd3 x 6( d2y)2 + Sdyddy 
(X—2)(Y~y) (Y-—y)dx—(X~a)dy 
2dedy d2yda—dedy =0 (41). 
3dxd2y + 3dyd2z, dByda— dexdy 
6d2xd2y + 4(ded3y + dyd32), d4yd2—d4xdy 
which is the equation of the conic of closest contact in general 
differentials. 
Equation (41) reduces to (38) when the independent variable 
is @ 
2. It is not difficult to extend the method of general 
differentials to the direct determination of the equation of the 
osculating a 
e equation of a curve passing through (2, y), (#,, y,) which 
eer to a parabola if (a, y) and (a, y1) coincide, is evidently of 
t 

Me/ (X—a@) (X—a) +ha/(¥-y) (Y—-H) 
=v/ (X=2) (yy) — (2-9) (@|—#)- 

Therefore, the equation of such a curve <= through any 
four points (#, y), (#1, Y,), (2 Y2)s (@s» Ys 
/(X—2) (X—2) YW (Y¥~y) (Y—yy) /(Y—y) (a - #)—(X-2)(yj— ) 
V (@2—#) (22-2) / (ay) (a—w) VW (va—¥) (21-2) —(ea—2) (1 —9) 
V/ (a3—2) (@3-2) M(vs—v) (s—w) YW (yay) (#1 —2) — (25-2) — 






Now if (a, y), ey yi (®, Yo), (xs, ¥s3) be four consecutive 
points on a curve, then m (5), 
J (a;—2) (@,— 2) = Vo (2d + dx) (da + dx) =./2 (dx+ $dx) 
/ (x,— 2) (2, —2,) =a/ (Bda+t 3d*x + dix) (2de+3d8x+ den) | 
= ~/6 ( 5. 8) 






VA (92 —y) @—2) — (4-2) (yy) =o Byda — Bady 
VS (¥3—y) (@—%) — (#3 -2) (41 ~Y) 
=/3(@yda—Pady) + (Pydz — dxdy) = /3V/ dadty —dydx 
x (142 dydz— sat ) 
® Giyda —dxdy ; 

r (43). 



