30 
ON A LOCUS DERIVED FROM TWO CONICS. 
[389 
so that 
A — 2lp, .. F = mr + nq, .. 
A' = 21'p’, .. F' = m’r' 4 n'q',.. 
(.BG — F 2 , ...) = — {mr —nq, np —Ir, Iq —mp) 2 , 
{B'C -F'\ ...) = - {mV - n'q', n'p - IV, l'q - m'pj, 
BC' + B'G — 2FF' = 2 {{mn — min) (qr' — qr) — {mr' — nq ) {m'r — n'q), .. 
and substituting these values the equation is 
{2k + l) 2 
a, /3, 7 
2 
a > /3, 7 
2 "{ 
a , /3 , 7 
a, /3, 7 
- 
a, /3, 7 
a, /3 , 7 
l, m, n 
1', ml, n 
l, m, n 
p, q, r 
l , m, n 
Ï, ml, nl 
p, q , r 
p, q , r' 
V, ml, n 
p, q', r' 
p', q, r' 
p, q , r 
) 2 = 0, 
which, if A, B, C denote 
a, 
13, 
7 
a , 
¡3, 
7 
y 
a, /3, 
7 
a, 
/3, 
7 
> 
a, 
/3, 
7 
<*> /3, 7 
l, 
m, 
n 
v, 
ml, 
n 
l, m, 
n 
p'> 
r 
l, 
m, 
n 
V, ml, n 
P> 
r 
P> 
<1 > 
r 
i , 771 , 
n 
P> 
r 
P'> 
4 > 
r' 
p, q , r 
respectively, (A + B + C = 0) is, in fact, the equation 
{2k + 1)*A*-(B- C) 2 = 0, 
or, what is the same thing, 
that is 
l- B l-° 
k - A 01 k ~ A ’ 
either of which expresses the anharmonic property of the points of a conic in the 
form given by the theorem ad quatuor lineas. 
Reverting to the case of two conics, then if these be referred to a set of con 
jugate axes, the equations will be 
aa?+ by* + cz*=0, 
ax 2 + b'y 2 4- dz 2 = 0, 
we have K = abc, K' — a'b'c', 
© = {be' 4- b'c) aa'x 2 4- {ca 4- c'a) bb'y 2 + {ab' + alb) ccz 2 , 
and the equation of the quartic curve is 
4 {2k + l) 2 abed b'c {ax 2 + by 2 + cz 2 ) {ax 2 4- b'y 2 + cz 2 ) 
— \{bc + b'c) aa'x 2 4- {cd + c'a) bb'y 2 + {ab' + ab) cc'z 2 } 2 = 0.