[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