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718] ADDITION TO MR GENESE’s NOTE ON THE THEORY OF ENVELOPES. 
It is to be remarked that writing in the equation of the parabola these values 
A, = 0, y = 0, b\ — ay = 0 successively, we find respectively 
x{oc — az) = 0, 
y{y — bz) = 0, 
(bx + ay) (bx + ay — abz) = 0 ; 
viz. in each case the parabola reduces itself to a pair of lines, one of the given 
lines and a line parallel thereto through the intersection of the other two lines; the 
parabola thus becomes a curve having a dp on the line at infinity. 
In the fourth case z = 0, the equation in A, y is (Ay + yxf = 0, giving a variable 
value \-r-fjb = — x + y; hence z — 0, the line at infinity is a proper envelope. 
The true geometrical result is that the envelope consists of the three points A, B, C, 
and the line at infinity; a point qua curve of the order 0 and class 1 is not represent 
able by a single equation in point-coordinates, and hence the peculiarity in the form of 
the analytical result.