MATHEMATICAL NOTES, 291 
directions of the asymptotes, and the coefficient of the next lower 
power of + which does not identically vanish for these values of 7: m, 
will, on being equated to zero, give the asymptotes. 
This also shews clearly the reason of the occasional failure of the 
common rule, when terms of the second highest dimension are want- 
ing, viz.: equate to zero the terms of the highest dimension. The 
rule succeeds when the expression of the highest dimensions consists 
of factors occurring singly, but may fail when the same factor occurs 
in it more than once. \, 
2. On a Reduction of Curves of the Second Order: 
In the modern system of analytical geometry, as pursued by 
Salmon, Puckle, and others, the curves of the second order, as 
represented by the general equation in Cartesian rectangular co- 
ordinates, are first separated into central and non-central, and the 
further reduction of the equation is then effected by transformation 
of coordinates, which is a rather long and troublesome process. It 
has occurred to me that this reduction might be simplified by follow- 
ing the course taken by Euclid with regard to the circle, namely, by 
seeking whether there exists a line (or lines) with regard to which 
the curve is symmetrical. For this purpose let us take the curves 
separately. 
I. Central curves, C? — AB is not zero, and the equation referred 
to the centre takes the form 
Ag? + By? + 2 Cry = F. 
Let the curve be cut by the line 
e—a 
y 
then we obtain a quadratic for the values of r at the points of sec- 
tion, by substituting for z, y, in the equation to the curve, and the 
coefficient of the simple power of r in this, is 
Ale + BmB + CIB + ma), 
and if this vanish, the values of » are equal and opposite, and (a, 6) 
will be the middle point of the chord of section. Now this condi- 
tion is 
(Al + Cm) a + (Bm + Cl) B= 0 vaceecsecsarvee (2) 
and if 7: m be given, the locus of .this equation is a straight line 
through the origin. 
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