351] 
ON CUBIC CONES AND CURVES. 
411 
On the Anharmonic Property of a Cubic Cone. Nos. 25 to 29. 
25. The property in question is the one already referred to, viz. the four tangent 
planes, or say the four tangents, to the cone from any line of the cone form a 
system the anharmonic ratios of which are constant. Taking the equations of the 
tangents to be 
p — aq = 0, p — bq — 0, p — cq = 0, p — dq = 0, 
T 2 
and writing for shortness m = 64 — , then the functions 
{a -b)(c- d), (a — c)(d — b), (a — d)(b — c), 
or say a, ¡3, y, on which the anharmonic ratios depend, are the roots of the equation 
t s — 12t + 2 fm = 0. The anharmonic ratios are ~ , — , -, -, —, \; hence forming 
p a y a y p 
the equation ^ = 0, and reducing by the conditions, 
a + ¡3 + y = 0, 
-P *ya -P a/3 = — 12, 
a/3y = — 2 Vm, 
.... . . /rxoCNlX 6y - , Vm ^P^Pl J -I 
this is found to be (^ 2 + ^ + l) + - F PA = 0, or we have y = —w ^ » and sub- 
VTO 0 
stituting this value in the equation y 3 — 127 p 2 fm = 0, we find 
* 
(^ 2 + s +1) 3 - + * +1) 4 - 3 — 2 - 43 ^ = 0, 
\ / \ mm 
which is 
m (S 2 P * + l) 3 - 432 ^ 2 (* + l) 2 = 0, 
or, what is the same thing, 
Q 2 + + l) 3 _ 432 _ 432 
that is 
S 2 (>+ l) 2 
(^ 2 P ^ + l) 3 
* 2 (* + l) 2 
m 
64- 
jr 2 > 
s* 
27 
4 1 
T 2 y 
64>S 3 ) 
and as a verification it may be remarked that, 6 being a root, the six roots are 
A 1 /1 I m 1 0 
°’ 6’ K h 1 + 0’ 1 + 0’ 
1 + 0 
of course the roots are all real or else all imaginary. 
52—2