PROBLEMS AND SOLUTIONS. 
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The equation of the required conic may be taken to be © = \U+yV= 0, and 
this being so, the equation of any conic passing through the points of intersection of 
the conic 0 = 0 and the given conic W=0, will be \U+yV+ vW= 0; and if v be 
properly determined, viz. by the equation 
Disct. (X.U + yV+ vW) = 0, 
which it will be observed is a cubic equation in (X, y, v), then \JJ + yV + vW = 0 will 
be the equation of a pair of the chords of intersection of the conics 0 = 0, W = 0. 
The chord which passes through the given point (a, ¡3, 7) may be taken to be one 
of the pair of chords; the pair of chords, regarded as a conic, then passes through 
the given point (a, /3, 7); or if U 0 , V 0 , W 0 are what U, V, W become on sub 
stituting therein the values (a, ¡3, 7) for the coordinates, we have 
Xt/ 0 + /xk r 0 + v TTjq = 0, 
which is a linear equation in (X, y, v); and combining it with the before-mentioned 
cubic equation, 
Disct. (X U 4- y V + v W) = 0, 
the two equations give the ratios (X : y : v), and the equation of the required conic 
is \U + yV=0. There are three systems of ratios X : y : v, and consequently three 
conics satisfying the conditions of the Question. 
Suppose that the conics U— 0, V=0, W = 0, have a common chord, then the. 
conics © = \U + yV= 0, W = 0, have this common chord, say the fixed chord; and they 
have besides another chord of intersection, say the proper chord, which is the line 
joining the two points of intersection not on the fixed chord. It follows that, in 
the equation \U + yV +vW = 0, v may be so determined that this equation shall 
represent the fixed and proper chords; the required value of v is one of those given 
by the before-mentioned cubic equation, which will then have a single rational factor 
of the form a\ + by + cv, and the value of v is that obtained by means of this factor, 
that is, by the equation «X + by + cv = 0. 
{The value in question may, however, be found independently of the cubic equation; 
thus, if the three conics have the common chord P = 0, then their equations may be 
taken to be TJ = 0, U+ PQ = 0, U + PR = 0 ; we have then ® = \U + y(U + PQ), and 
the value of v is at once seen to be v = — (X + y), for then 
\U+yV+vW=\U + y(U+PQ)-(\ + y)(U+PR) = 0, 
that is, P [yQ — (\ + y)R] = 0, which is the conic made up of the fixed chord P = 0 
and the proper chord yQ — (X + y) R — 0.} 
But returning to the equations U = 0, V = 0, TT = 0, the value of v is given by 
the equation aX + by + cv = 0, obtained by equating to zero the rational factor of the 
cubic equation. Suppose now that the proper chord passes through the given point 
(a, /3, 7), then, as before, the equation \U + yV + vW = 0 is satisfied by these values 
c. v. 74