[330 
331] 
131 
, and 6 as the 
n through the 
v axes through 
le plane of ac, 
b), the equation 
= 0. 
331. 
ANALYTICAL THEOREM RELATING TO THE FOUR CONICS 
INSCRIBED IN THE SAME CONIC AND PASSING THROUGH 
THE SAME THREE POINTS. 
[From the Philosophical Magazine, vol. xxvn. (1864), pp. 42, 43.] 
Imagine the four conics determined, and, selecting at pleasure any three of them, 
let their chords of contact with the given conic be taken for the axes of coordinates, 
or lines x = 0, y — 0, z = 0; then, taking for the equation of the given conic 
U = (a, b, c, f g, K$x, y, zf = 0, 
the equations of the selected three conics must be of the form TJ + lx 2 =0, U+ my 2 = 0, 
U + nz 2 = 0, where l, m, n are to be determined in such manner that these conics 
may have three common points; the resulting values of l, m, n, and of the coordinates 
of the three common points, that is, the three given points, will of course be functions 
of the coefficients (a, b, c, f g, h); and the equation of the fourth conic will be of the 
form U + (ix +jy + kz) 2 = 0. 
There is no difficulty in carrying out the investigation: it is found that the coordi 
nates of the given points must be taken to be 
(- f g, h); (J,-g,h)\ (f 9> ~ h ) 
respectively, and that, writing as usual 
K = abc - af - bg 2 - cli 2 + 2 fgh, 
17—2