ON THE CUBIC CENTRES OF A LINE WITH RESPECT TO 
THREE LINES AND A LINE. 
[From the Philosophical Magazine, vol. xxir. (1861), pp. 433—436.] 
On referring to my Note on this subject {Phil. Mag. vol. xx. pp. 418—423, 
1860 [257]), it will be seen that the cubic centres of the line 
\x + gy + vz = 0 
in relation to the lines x = 0, y = 0,z = 0, and the line x + y + z = 0, are determined 
by the equations 
111 
x : y : z = -x—- : a : —, 
0 + X 0 + g 0 + v 
where 6 is a root of the cubic equation 
1 1 1 2 
0 + X Jr 0 + g~ >r 0 + v 0 
or as it may also be written, 
0 s — 0 {gv + vX + Xg) — 2Xgv = 0. 
Two of the centres will coincide if the equation for 0 has equal roots; and this will 
be the case if 
X~*+g~* + v ^ — 0, 
or, what is the same thing, if X, g, v = a~ 3 , h~ 3 , c~ 3 , where a + b + c = 0. In fact, if 
a + h + c = 0, then a 3 + h 3 + c 3 = 3abc, and the equation in 0 becomes