570 
PROBLEMS AND SOLUTIONS. 
[705 
the same point; and similarly the lines PR, Q8, pr, qs, and the lines PS, QR, ps, qr): 
it is required to show that a conic may be drawn, passing through the points p, q, r, s 
and touched at these points by the lines pP, qQ, rR, sS, respectively. 
Solution by the Proposer. 
Taking the centres for the vertices of the fundamental triangle, the equation of 
the given conic may be taken to be x 2 + y 2 + z 2 = 0; and then the coordinates of P, 
Q, R, S to be (A, B, C), (A, - B, C), (A, B, —G), (A, —B, —G) respectively, where 
A 2 + B 2 + C 2 = 0; and those of p, q, r, s to be (a, /3, 7), (a, — /3, 7), (a, /3, — 7), 
(a, — ft, — 7) respectively, where ot 2 + /3 2 + y 2 — 0. The required conic, assuming it to 
exist, will be given by an equation of the form lx 2 + my 2 + nz 2 = 0. This must pass 
through the point (ot, /3, 7), and the tangent at this point must be 
x (By — (7/3) + y ((7a — Ay) 4- z (A/3 — Ba.) = 0 ; 
that is, we must have loC- + m/3 2 + ny 2 = 0, and 
lot. : m/3 : ny = By—C/3 : Got —Ay : A/3 — Ba. 
The first condition is obviously included in the second; and the second condition 
remains unaltered if we reverse the signs of B, /3, or of G, 7, or of B, /3 and (7, 7. 
Hence the conic passing through p, and touched at this point by pP, will also pass 
through the points q, r, s, and be touched at these points by the lines qQ, rR, sS, 
respectively ; that is, the equation of the required conic is 
By - Gß 
a 
or, what is the same thing, 
ßyx 2 , y ay 2 , otßz 2 — 0. 
A , B , G 
a , ß , 7 
[Vol. xv., January to June, 1871, pp. 17—20.] 
3206. (Proposed by Professor Cayley.)—In how many geometrically distinct ways 
can nine points lie in nine lines, each through three points ? 
3278. (Proposed by Professor Cayley.)—It is required, with nine numbers each 
taken three times, to form nine triads containing twenty-seven distinct duads (or, what 
is the same thing, no duad twice), and to find in how many essentially distinct ways 
this can be done.