584 
PROBLEMS AND SOLUTIONS. 
[383 
or what is the same thing 
{varj + p (7I — a^)} 2 + 2v (pa + q<y) + 2p (ra + sy) + t = 0 ; 
where p, q, r, s, t are given functions of (f, tj, £). 
I write for greater convenience 
V = J[’ ^ = y > a = W, 7 = Z, 
(so that the quantities to be determined will be the ratios X : Y : Z : W); the 
foregoing equation then becomes 
( W 1 1 2 9 9 
U y + Yw - W} + y (pw + iZ)+ f (>■ W + sZ) + t = 0, 
or what is the same thing 
{ V YW+X (%Z- £W)Y + 2XY- (pW + qZ) + 2X 2 Y (rW+ sZ) + tX-Y> = 0. 
Hence, considering in place of the line %x+ r\y + £z = 0, the three given lines 
%iX + r] x y -f £iZ = 0, £># + y 2 y + % 2 z = 0, ^ 3 x + y 3 y + £ 3 z = 0 successively, we have the three 
equations 
{ Vl YW + X (&Z - & W)Y + 2IP ( Pl W + q x Z) + 2X 2 F (r a W + s x Z) + t v X 2 P = 0, 
[r). 2 YW + &c. } 2 + &c. = 0, 
{%FTF + &c. } 2 + &c. = 0; 
which, treating X, F, Z, W as the coordinates of a point in space, are each of them 
the equation of a quartic surface having the line (X = 0, F = 0) for a cuspidal line. 
The required values of X, F, Z, W are the coordinates of a point of intersection of 
the three surfaces, and these being found the equation of the conic satisfying the 
conditions of the question is 
(a, b, 0,/, g, K§cc, y, zf+ 2 (Wx + Zz) (JL + = 0. 
As to the intersection of surfaces having a common line, see Salmon’s Solid 
Geometry, p. 257; but the case of a cuspidal line not having been hitherto discussed, 
I am not able to say now how many points of intersection there are of the three 
surfaces, nor consequently what is the number of the solutions of the question in 
hand. It of course appears that 64 is a superior limit. 
(¡3) Through four given points to draw a conic such that one of its chords of 
intersection with a given conic shall pass through a given point. 
Let the four points be given as the intersections of the conics ¿7=0, V=0, and 
let W — 0 be the equation of the given conic, (a, /3, 7) the coordinates of the given 
point.