1893.] of circles ; and other systems of two tetrads. 55 



of circles ; and it is possible out of the 70 tetrads to select (and 

 that in G ways) a tetrad of blue circles, such that there exists 

 a fourth red circle touching each of these four blue circles. We 

 have thus a system depending upon 3 arbitrary circles, and for 

 which, therefore, the capacity is = 9. It is (as is known) possible, 

 in quite a 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 O. We have then four circles, each 

 having its centre in the line O 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 conies, each of them having for foci the centres of 

 these circles : the line £1 meets each of these conies in two points, 

 and there are thus on the line II 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 II, 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 H, 

 and touching each of the red circles 1, 2 and 3. And it is more- 

 over clear that taking the red circle 4 equal to 1 and situate 

 symmetrically therewith in regard to the line H, 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 

 H, that is the line H 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 II becomes thus a circle H, orthotomic to 

 each of the blue circles. 



Starting ah initio, we have here at pleasure the red circles 

 1, 2, 3 : the circle H 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 H. We can then find 4 circles each of 



