"TT
113]
53
113.
NOTE ON THE GEOMETRICAL REPRESENTATION OF THE
INTEGRAL J dx + J(x + a) (x + b) (x + c).
[From the Philosophical Magazine, vol. v. (1853), pp. 281—284.]
The equation of a conic passing through the points of intersection of the conics
x 2 + y 2 + z 2 — 0,
is of the form
ax 2 + by 2 + cz 2 = 0,
w (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0,
where w is an arbitrary parameter. Suppose that the conic touches a given line, we
have for the determination of w a quadratic equation, the roots of which may be
considered as parameters for determining the line in question. Let one of the values
of w be considered as equal to a constant quantity k, the line is always a tangent
to the conic
k (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 = 0;
and taking w = p for the other value of w, p is a parameter determining the parti
cular tangent, or, what is the same thing, determining the point of contact of this
tangent.
The equation of the tangent is easily seen to be
x VT+l; \/a+p + y \/c — a*/b + k^/b+p + z'/a-b\/c + k\/c+p=0;
suppose that the tangent meets the conic a? + y 2 + z-= 0 (which is oi course the
conic corresponding to w = co) in the points P, P, and let 0, co be the parameters
of the point P, and 6', the parameters of the point P', i.e. (repeating the defini-