Vol. IV, No. 4.] A General Theory of Osculating Oonics. 177 
N.S. 
substituting (43) in (42) and simplifying, we have 
X-2 Y-y 3,4/2(dad*y —dyd*x)t/ (Y—y) dxe—(X—a)dy 
dz dy 3(dxd*y - dyd*r) 
de diy (dad'y —dyd®z) 

=0 (44). 
or, (Y—y) { de(d8ydz —d’zdy) —3d*a(d*ydx—dady) } 
—(X=2) { dy(d'ydze —Bady) —3%y(d?ydz — d’xdy) J 
=3,/2(d®ydx —d®xdy)2/ (Y—y) dx—(X—a)dy (45). 
h is the equation of the osculating parabola in general 
——— It reduces to (21) if « be the independent variable. 
rom (45) it is evident that the equation of the line of 
centres in general differentials is 
(Y-y) { dx(dydx —d®xdy) —3d*x(d*ydx— d*xdy) } 
=(X—-«) ( dy(@ydz —d'zdy) —3d*y(d*ydx -—-d*zdy)} (46). 

the conic of closest contact is stationary. We may determine 
he condition that any six points (a, y), (@), 7), (a. Yo), (@s» ¥3)> 
(%y Y4), (#5, ¥s,) may lie on a conic is, evidently, : 
(w2—2) (wg—21) (ye—-y) (ve-m) (Ya—m1) (22-2) 
(wg— 2) (eg—21) (ys—y) (vs—vi) (ys—m) (23-2) 
(%—#) (%—2) (ys—y) (ys—m1)  (va- 41) (4-2) 
(ap—2) (v§—2) (y5—y) (vs—41) (ys—m1) (75—2) 
(ya— y)(#1—#) —(yi—y)(za— 2) 
(yg—y)(#1— 2) — (yi —y)(@3—2) 
(ys—y) (21-2) —(1— 9 (24-2) 
(ys—y) (21 —2)—(yi—ye5—2) 
Now if (#, y), (%1, ¥1), (ar Yo)» (35 Ya)s (as Ya)s (59 Ys) be six 
consecutive points on a curve, then as in (5), 
@=e2+de w=a+2de+@e x,=e+3de 4302+ Pau } 
&,= 2+ 4da+ bd*x + 4d5x + da t 
%,= 2+ Sdx + 10d4x + 10d3x + 5dte + dx H 
with corresponding expressions for ¥, Y2) Y¥3 Ya Ys =) 
=0 (47). 
(48). 
n substituting (48) in (47), we have, after simplification 
of the determinant by adding to the second row, the first row 
d d 
the first row multiplied by —10, and ultimately neglecting all 
infinitesimals of higher orders, 
