NOTE ON CERTAIN SYSTEMS OF CIRCLES. 
113 
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183] 
To obtain a distinct idea of the methods made use of in Gaultier’s “ Mémoire 
sur les moyens généraux de construire graphiquement un cercle déterminé par trois con 
ditions,” {Jour. Ecole Polyt. t. ix. [1813], p. 124), and in Steiner’s “Geometrische Betrach- 
tungen,” Grelle, t. I. [1826], p. 161 ; it should be remarked that both of these geometers, 
confining as they do their attention to real circles, do not consider the section circle 
of an exterior point, or the taction circle of an interior point. The taction circle of 
an exterior point, or the section circle of an interior point, is Gaultier’s “Cercle radical,” 
and Steiner’s “ Potenzkreis,” and Steiner also speaks of the radius of this circle as the 
“ Potenz ” of its centre in relation to the given circle. The nature of the Cercle 
radical, or Potenzkreis, (i.e. whether it is a taction circle or a section circle) is of 
course determined as soon as it is known whether the centre is an exterior or an 
interior point, and Gaultier distinguishes the two cases as the “radical réciproque ” and 
the “radical simple,” and in like manner Steiner speaks of the Potenz as being 
“ auszerlich ” or “ innerlich.” Again, for two circles and for a given centre of similitude 
Gaultier and Steiner employ the tactaction circle or the sectaction circle, whichever 
of them is real, Gaultier without giving any distinctive appellation to the circle in 
question, Steiner calling it the Potenzkreis of the two circles, and in particular the 
“auszere Potenzkreis” or the “innere Potenzkreis,” according as it has for centre the 
centre of direct similitude or the centre of inverse similitude. 
The preceding properties of circles are of course at once extended to conics 
passing each of them through the same two points ; it is I think worth while to 
notice what the analogue is of a pair of concentric orthotomic circles. If the fixed 
points are I, J and if the point corresponding to the centre is K, then the conics 
are of course conics touching the lines Kl, KJ in the points I, J, and, one of the 
conics being given, the other is to be determined. It is easily seen that if an arbitrary 
line through / meets the conics in P, P' and the line KJ in M, then the points 
I, M, P, P' are a harmonic range, and this condition gives the construction of 
the second conic ; it of course follows that an arbitrary line through J meets the 
conics in points Q, Q' and the line KI in a point N such that the points J, N, Q, Q' 
are also a harmonic range. The two conics in question may be termed “inscribed 
harmonics” each of the other. 
Addition. The equation of the tactaction circle, corresponding to the centre of 
direct (or inverse) similitude, of two given circles, may be found as follows: 
Let the equations of the given circles be 
(x — a) 2 + {y — /3) 2 = c 2 , 
(x - ay + (y- pj = c' 2 , 
then the coordinates of the centre of direct similitude are 
ac — a'c fic — fi'c 
c' — c' ’ c —c 5 
which are therefore the coordinates of the centre of the tactaction circle ; and the 
equation of this circle is of the form 
\ [{x - af + {y- Pf - c 2 ] + (1 - X) [{x - a') 2 + (y - P'f - c 2 ] = 0, 
C. III. 
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