[705
705]
PROBLEMS AND SOLUTIONS.
569
(so that a+h—g= 0, &c., a + b + c = 0, af+ bg + ch = 0), the six points lie in each of
the plane-pairs
x {hy — gz + aw) = 0, y {— hx +fz + bw) = 0,
z (gx — fy + cw) =0, w {— ax — by — cz ) = 0.
We cannot take these as the four quadrics, on account of the identical equation
0 = 0, which is obtained by adding the four equations; but we may take the first
three of them for three of the quadrics, and for the fourth quadric the cone, vertex
(0, 0, 0, 1), which passes through the other five points; viz. this is
thus
P = x {hy — gz -I- aw), Q = y {— hx +/z + bw),
R = z (gx —fy+ cw), S = aayz + bßzx + cyxy ;
and we equate to zero the determinant formed with the derived functions of P, Q, R, S
in regard to the coordinates (x, y, z, w) respectively. If, for a moment, we write
A, B, G to denote bg — cli, ch — af, of —bg respectively, it is easily found that the
term containing d x S is
(b(3z + cyy) x (— agh, bhf, cfg, abc, —a/ 2 , —gB, hC, aA, b 2 g, -c 2 h\x, y, z, tv) 2 :
the terms containing d y S and d z S are derived from this by a mere cyclical interchange
of the letters (x, y, z), {A, B, C), (a, b, c), and (f g, h). Collecting and reducing, it
is found that the whole equation divides by 2abc; and that, omitting this factor,
the result is
that
tance
ircles
idard
ayz (aw 2 — hx 2 ) + fxw (ßz 2 — yy 2 ) '
-I- bzx (ßw 2 — By 2 ) + gyw (yx 2 — az 2 ) 1 = 0,
+ cxy (yw 2 — hz 2 ) + hzw {ay 2 — ßx 2 )
which, substituting for a, b, c, f g, li their values, is the required form.
line
acted
If, in the equation, we write for instance x = 0, the equation becomes
ayzw {hy — gz + aw) = 0;
or, the section by the plane is made up of four lines. Calling the given points
1, 2, 3, 4, 5, 6, it thus appears that the surface contains the fifteen lines 12, 13, ..., 56,
and also the ten lines 123.456, &c.; in all twenty-five lines. Moreover, since the
surface contains the lines 12, 13, 14, 15, 16, it is clear that the point 1 is a node
(conical point) on the surface; and the like as to the points 2, 3, 4, 5, 6.
[Voi. xiv., pp. 104, 105.]
3249. (Proposed by Professor Cayley.)—Given on a given conic two quadrangles
PQRS and pqrs, having the same centres, and such that P, p; Q, q; R, r; S, s
are the corresponding vertices (that is, the four lines PQ, RS, pq, rs all pass through
C. X.
72