588 
DR, J. CASEY OX CYCLIDES AXD SPHERO-QUAETICS. 
of the result equated to zero gives, as is easily seen, 
(1 s ")/A + 2(1 - U)k + (1 - S') = 0, (9) 
where S', S" are the results of substituting the coordinates of the poles of the planes 
A and B in S, and B the result of substituting the coordinates of the pole of A in B. 
We should arrive at the same result if we had taken the equations of the two quadrics 
under the form S^fi-A and S* + B. But if we had worked with S^+A and SAB, we 
should get 
(1 - S")/A + 2(1 + R)£ + (1 - S') (10) 
8. As the equations (9) and (10) are of the second degree in Jc, we see that, through 
each conic of intersection of S — A 2 and S — B 2 , there pass two cones circumscribed to S. 
The equations of these cones are obtained by eliminating Jc between — Afi-&(S J — B), 
and the two equations (9) and (10). They are : 
(1 — S")(S> — A) 2 — 2(1-R)(S*- A)(S*- B)+(1-S')(S*- B) 2 , . . (11) 
(l-S'O(Si-A) 2 — 2(l + K)(S^-A)(Si+B) + (l — S')(S^+B) 2 . . . (12) 
These cones correspond to the limiting points of two spheres, as these latter are evidently 
imaginary cones passing through the circle of intersection of the two spheres and cir- 
cumscribed to the imaginary circle at infinity. 
9. If we put 
1— R=V(1— S')(l-S") cos 0, 
1 + R= X /(1-S')(1-S") cos <p, 
the ratio of the roots of equation (9) is and of equation (10) e 2<pv/_1 . Now if 
7T 
S = - the ratio of the roots is negative unity, and we have an harmonic pencil of four 
planes, namely the planes A, B, and the planes passing through the intersection of the 
planes A, B, and which are also planes of contact of the cones of article (8) with the 
quadric (S) ; in other words, the poles of A and B and the vertices of the cones form an 
harmonic range of points. When two quadrics, then, are connected by the relation 
ldtH=0, (13) 
I shall, by an extension of a known term, say that they cut orthogonally or harmonically. 
10. It is easy to see that, being given by its general equation, a quadric S, and two 
planes 'kx-\-yjy-\-vz-\-gw, the result of substituting the coordinates of 
the pole with respect to the quadric of one of the planes in the equation of the other, 
multiplied by the discriminant of the quadric, is equal to the determinant : 
a, 
n, 
TO, 
lb 
n , 
l, 
fA 
TO, 
l, 
c , 
r, 
iA 
r , 
d, 
§ ’ 
g', 
0; 
