74 
ON THE CUBIC CENTRES OF A LINE 
[315 
that is 
(abed) 3 — 3 (a bed) — 2=0, 
which is 
(abed + l) 2 (abed — 2) = 0 ; 
so that the values of d are -7- , -y . 
abc abc 
First, if d = — 7-, then x, y. z will be the coordinates of the double centre. And 
abc J 
we have 
8 ■ +x - h ~ ¿0 - me (26c - 2 ° !) =2- & s - <0 ; 
or putting for shortness □ = a 2 4- b'~ + c 2 , 
a . _ 1 n _ 3 □ 
2 a 3 bc ’ abc' 6a 2 ’ 
with similar values for d + /u, 0 4- z>. 
and Ave may therefore write 
But - , - , - are proportional to d + A, d + ¡i, d + v ; 
x y z 
P = _D P □ P = D. 
¿r 6a 2 ’ y 66 2 ’2 6c 2 ’ 
whence, in virtue of the equation a + b + c = 0, we have for the locus of the double 
centre, 
*Jx + *Jy+kz = 0; 
or this locus is a conic touching the lines x = 0, y = 0, 2 = 0 harmonically in respect 
to the line x + y + 2 = 0, a result which was obtained somewhat differently in the 
paper above referred to. 
have 
Next, if Q — x , V> z will be the coordinates of the single centre. And we now 
* +x=+ Js = 2k (2bc ~ +6a ‘ }= mc ( ~ a+6a?) —k ■ 
with similar values for d + ¡i, d + v. 
and we may therefore write 
But -, - , - are proportional to d + \, d + /x, d + v, 
x y z 
P □ — 6n 2 P 
□ - 6Ò 2 P 
□ — 6c 2 
X 6tt 2 ’ y 
66 2 ’2 
6c 2 ’ 
from which equations, and the equation 
to be eliminated. I at first effected 
a + b + c = 0, the quantities P, 
the elimination as follows: viz., 
a, b, c have 
writing the 
equations under the form 
x 6a 2 y 
6Ò 2 2 
6c 2