426 
ON A SYSTEM OF TWO TETRADS OF CIRCLES ; 
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different manner, out of the 70 tetrads to select (and that in 8 ways) a tetrad of 
blue circles such that there exists a fourth red circle touching each of these four 
blue circles—but the present paper relates exclusively to the first-mentioned 6 tetrads 
and not to these 8 tetrads. 
3. I consider, in the first instance, a particular case in which the three red 
circles are not all of them arbitrary, but have a capacity 9 — 1, = 8; and pass from 
this to the general case where the capacity is = 9. Calling the red circles 1, 2, 3 
and 4; I start with the circles 1, and 2 arbitrary, and 3 a circle equal to 2: the 
radical axis, or common chord, of the circles 2 and 3 is thus a line bisecting at 
right angles the line joining the centres of the circles 2 and 3, say this is the 
line 12. We have then four circles, each having its centre in the line 12 and 
touching the circles 1 and 2: in fact, the locus of the centre of a circle touching 
the circles 1 and 2 is a pair of conics, each of them having for foci the centres of 
these circles: the line 12 meets each of these conics in two points, and there are 
thus on the line 12 four points, each of them the centre of a circle touching the 
circles 1 and 2. But the equal circles 2 and 3 are symmetrically situate in regard 
to the line 12; and it is obvious that the four circles, having their centres on the 
line 12, will each of them also touch the circle 3; we have thus the four blue 
circles, each of them with its centre on the line i2 and touching each of the red 
circles 1, 2 and 3. And it is moreover clear that, taking the red circle 4 equal to 
1 and situate symmetrically therewith in regard to the line 12, then this circle 4 
will touch each of the blue circles: so that we have here the four blue circles, each 
of them touching the four red circles. As already mentioned, the blue circles have 
their centre on the line 12, that is, the line il is a common orthotomic of the 
four blue circles. 
4. By inverting in regard to an arbitrary circle we pass to the general case; 
the line 12 becomes thus a circle 12, orthotomic to each of the blue circles. 
Starting ab initio, we have here at pleasure the red circles 1, 2, 3: the circle 
12 is a circle having for centre a centre of symmetry of the circles 2 and 3, and 
passing through the points of intersection (real or imaginary) of these two circles; 
the circles 2 and 3 are thus the inverses (or say the images) each of the other in 
regard to the circle 12. We can then find 4 circles each of them orthotomic to 12, 
and touching the circles 1 and 2: but a circle orthotomic to 12 is its own inverse 
or image in regard to 12; and it will thus touch the circle 3 which is the image 
of 2 in regard to 12. We have thus the four blue circles each of them touching 
the red circles 1, 2 and 3; and then, taking the red circle 4 as the inverse or 
image of 1 in regard to 12, this circle 4 will also touch each of the blue circles. 
Thus starting with the arbitrary red circles 1, 2, 3, we find the four blue circles 
and the remaining red circle 4, such that each of the blue circles touches each of 
the red circles. Since in the construction we group together at pleasure the two 
circles 2, 3 (out of the three circles 1, 2, 3) and use at pleasure either of the two 
centres of symmetry, it appears that the number of ways in which the figure might 
have been completed is = 6.