362 
[56 
56. 
DEMONSTRATION OF A GEOMETRICAL THEOREM OF JACOBI’S. 
[From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 48—49.] 
The theorem in question (Crelle, t. xii. [1834] p. 137) may be thus stated: 
“If a cone be circumscribed about a surface of the second order, the focal lines 
of the cone are generating lines of a surface of the second order confocal to the given 
surface and which passes through the vertex of the cone.” 
Let (a, /3, 7) be the coordinates of the given point, 
X 1 y 2 Z 2 , 
a 2+ 6 2 + c 2 
the equation to the given surface. The equation of the circumscribed cone referred to 
its vertex is 
X 2 y- Z- , , 
-2+1-2+72 Lj + 
6- c 2 
ax 
72 + 
ßy_ + yzy_ 
b 2 c 2 
0, 
whence it is easily seen that the equation of the supplementary cone (i.e. the cone 
generated by lines through the vertex at right angles to the tangent planes of the 
cone in question) is 
(ax + fiy + 7zf — a 2 « 2 — b 2 y 2 — c 2 z 2 = 0. ( x ) 
Suppose we have identically 
(ax+ /3y + yzf - a 2 x 2 — b 2 y 2 — c 2 z 2 — h(x? + y 2 + z 2 ) = (lx + my + nz) (I'x + my + nz); 
lx 4- my + nz = 0 will determine the direction of one of the cyclic planes of the supple 
mentary cone, and hence taking the centre for the origin the equations of the focal 
lines of the circumscribed cone are 
x — a _y — /3 z — 7 
l m n 
1 Ax 2 + By 2 + Cz 2 + 2Fyz + 2Gxz + 2Hxy = 0 being the equation of the first cone, that of the supplementary 
cone is ( ^iX 2 -\-^y 2 + i!iz 2 + ‘lSilz + 2<&zx + 2ysixy-0, these letters [denoting the inverse coefficients BC-F 2 , &c.].