493] 
31 
493. 
ON EVOLUTES AND PARALLEL CURVES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 183—200.] 
In abstract geometry we have a conic called the Absolute; lines which are 
harmonics of each other in regard to the absolute, or, what is the same thing, which 
are such that each contains the pole of the other in regard to the absolute, are said 
to be at right angles. Similarly, points which are harmonics of each other in regard 
to the absolute, or, what is the same thing, which are such that each lies in the 
polar of the other, are said to be quadrantal. 
A conic having double contact with the absolute is said to be a circle; the inter 
section of the two common tangents is the centre of the circle ; the line joining the 
two points of contact, or chord of contact, is the axis of the circle. 
Taking as a definition of equidistance that the points of a circle are equidistant 
from the centre, we arrive at the notion of distance generally, and we can thence 
pass down to that of equal circles; but the notion of equal circles may be established 
descriptively in a more simple manner : 
Any two circles have an axis of symmetry, viz. this is the line joining their 
centres; and they have a centre of homology, viz. this is the intersection of their 
axes. They intersect in four points, lying in pairs on two lines through the centre of 
homology: they have also four tangents meeting in pairs in two points on the axis 
of symmetry. Now if the two lines through the centre of homology are harmonically 
related to the two axes, or, what is the same thing, if the two points on the axis 
of symmetry are harmonically related to the two centres, then the circles are equal. 
Circles which are equal to the same circle are equal to each other, and the entire 
series of circles which are equal to a given circle, are said to be a system of circles of 
constant magnitude. 
Starting from these general considerations, I pass to the question of evolutes and 
parallel curves: it will be understood that everything—lines at right angles, circles, 
poles, polars, reciprocal curves, &c.—refers to the absolute.