"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-