183] 
111 
183. 
NOTE ON CERTAIN SYSTEMS OF CIRCLES. 
[From the Quarterly Mathematical Journal, vol. II. (1858), pp. 83—88.] 
It will be convenient to remark at the outset that two concentric circles, the 
radii of which are in the ratio of 1 : i (i being as usual the imaginary unit), are 
orthotomic^), and that the most convenient quasi representation of a circle, the centre 
of which is real and the radius a pure imaginary quantity, is by means of the con 
centric orthotomic circle. This being premised consider a circle and a point G. 
The points of contact of the tangents through G to the circle may be termed the 
taction points; the points where the chord through G perpendicular to the line joining 
C with the centre meets the circle, may be termed the section points. It is clear 
that, for an exterior point, the taction points are real and the section points imaginary, 
while, for an interior point, the section points are real and the taction points 
imaginary. A circle having C for its centre and passing through the taction points 
(in fact the orthotomic circle having G for its centre) is said to be the taction circle. 
A circle having G for its centre and passing through the section points is said to be 
the section circle. Of course for an exterior point the taction circle is real and the 
section circle imaginary; while for an interior point the taction circle is imaginary 
and the section circle is real. It is proper also to remark that the taction circle 
and the section circle are concentric orthotomic circles. 
Passing now to the case of two systems of orthotomic circles, let MM', NN' be 
lines at right angles to each other intersecting in R, and let M, M be real or pure 
imaginary points on the line MM', equidistant from R. Imagine a system of circles, 
1 Two concentric circles are, it is well known, conics having a double contact at infinity, and it appears 
at first sight difficult to reconcile with this, the idea of two particular concentric circles being orthotomic. 
The explanation is that any two lines through a circular point at infinity may be considered as being at 
right angles to each other, and therefore any line through a circular point at infinity may be considered 
as being at right angles to itself. The two concentric circles in question have, in fact, at each circular 
point at infinity a common tangent, but this common tangent must be considered as being at right angles 
to itself. The paradox disappears entirely upon a homographic deformation of the figure; two lines KL, 
KM are then defined to be at right angles when joining K with the fixed points I, J, the four lines KL, 
KM, KI, KJ are a harmonic pencil; but when K coincides with I, then KI is indeterminate and may be 
taken to be the fourth harmonic of the pencil, i.e. any two lines IL, IM through the point I may be 
considered as being at right angles.