586 
PROBLEMS AND SOLUTIONS. 
[383 
of the coordinates, or we have Xi7 0 + /AF 0 + vW i) = 0; which, with the before-mentioned 
linear equation aX + by + cv = 0, determines the ratios \ : y : v; and the required conic 
is \U+yV=0‘, there is, then, in the present case only one conic satisfying the 
conditions of the Question. 
(7) Through two given points to draw a circle such that its chord of intersection 
with a given circle shall pass through a given point. 
The foregoing discussion of the case of three conics having a common chord is 
of course directly applicable to the present question, the common chord being the line 
infinity; it is therefore sufficient to give the final analytical result; viz., if the given 
points are y = 0, x—±l, and if the given circle is x 2 + y 2 + c+ 2fy + 2gx= 0, and the 
point through which passes the chord is x = a, y = /3, then the equation of the required 
circle is 
x 2 + y 2 — 1 + i (2ga + 2/J3 + 1 + c) y = 0. 
The equation of the chord of intersection is, in fact, 
1 +c-~(2ga + 2f^+l + c)y + 2gx + 2fy = 0, 
which is satisfied, as it should be, by x = a, y = (3. 
But the geometrical solution is even more simple. Let A, B, be the given points, 
G the point through which passes the chord of intersection; then, joining G, A, and 
taking on this line a point H such that GA . GH is equal to the square on the 
tangential distance of G from the given circle, it is at once seen that any circle 
through A y H is such that its chord of intersection with the given circle passes 
through G; hence the required circle is that drawn through the three points A, H, B. 
[Vol. III. January to July, 1865, p. 29.] 
1607. (Proposed by Professor Cayley.)—In a given cubic curve to inscribe a 
triangle such that the three sides shall pass respectively through three given points on 
the curve. 
[Vol. ill. pp. 60—63.] 
1647. (Proposed by Professor Cayley.)—Find the locus of the foci of an ellipse 
of given major axis, passing through three given points. 
{In connexion with the problem the Proposer remarks as follows : 
Let A, B, G be the given points; take P an arbitrary point (not in general in 
the plane of the three given points), then we may find a point Q (not in general