208 
ON THE CONIC OF FIVE-POINTIC 
[261 
will be the equation of a conic having an ordinary (two-pointic) contact with the curve 
at the point (x, y, z). In fact the equation DU = 0 is that of the tangent at the point 
in question, and the equation D 2 U = 0 is that of the penultimate polar (or polar conic) 
of the point, which conic is touched by the tangent; the assumed equation represents 
therefore a conic having an ordinary (two-pointic) contact with the polar conic, and 
therefore with the curve. It may be added that the two conics intersect besides in 
a pair of points, and that the line joining these, or common chord of the two conics, 
is the line represented by the equation II = 0; and this being so, the constants (a, b, c) 
of the line II = 0 can be so determined as to give rise to a five-pointic contact. 
3. Consider the coordinates of a point of the curve as functions of a single 
variable parameter; then for the present purpose the coordinates of a point consecutive 
to (x, y, z) may be taken to be 
x + dx + \ d 2 x + d’x + -gL d*x, 
y + dy + ^d 2 y + ^ dhy + £ d% 
z + dz +%d-z + £ d s z + d*z, 
values which, substituted for X, F, Z, must satisfy the equations 
T = 0, D 2 U — II. DU = 0. 
4. I write for shortness 
= d xd x + dyd y + dzd 2 , 
0o = drx d x + dhy dy + d 2 z d z , 
0 3 = d 3 x d x + dhy d y + d 3 z d z> 
0 4 = d 4 x d x + dhy d y + d 4 z d z , 
then the consecutive value of T is 
exp. (0j + £ 0 2 + £ 0 3 + 0 4 ) U 
(Read exp. z, exponential of z, = e 2 ), which is 
- (l + ai + w + tff+M 4 )' 
x(l 
+ |0 2 
+ i a 2 2 ) 
x(l 
+ Fa 
* ) 
x(l 
+ 21^4) 
= 1 + 0! + ^01 2 + ¿0i' ! 
+ F-' + FF2 
+ Wd, 
+ 
+Fs 
+ h 9 3 
+ ^j04 
= 
u 
+ 
djj 
+ \ (0i 2 + 0 2 ) U 
+ i (d 1 3 +3d 1 d, +d 3 ) U 
+ -gif (0i 4 + G0J-0.. + 40>0 3 + 30 2 2 + 0 4 ) U,