ON A SYSTEM OF TWO TETRADS OF CIRCLES; AND OTHER 
SYSTEMS OF TWO TETRADS, 
[From the Proceedings of the Cambridge Philosophical Society, vol. vm. (1893), 
pp. 54—59.] 
The investigations of the present paper were suggested to me by Mr Orr’s 
paper, “The Contacts of certain Systems of Circles,” Proc. Camb. Phil. Soc., vol. vii. 
1. It is possible to find in piano two tetrads of circles, or say four red circles 
and four blue circles, such that each red circle touches each blue circle: in fact, 
counting the constants, a circle depends upon 3 constants, or say it has a capacity 
= 3; the capacity of the eight circles is thus = 24; and the postulation or number 
of conditions to be satisfied is = 16: the resulting capacity of the system is thus 
primd facie, 16 — 24 = 8. It will, however, appear that, in the system considered, the 
true value is = 9. 
2. The primd facie value of the capacity being = 8, we are not at liberty to 
assume at pleasure three circles of the system. And, in fact, assuming at pleasure 
say 3 red circles, then touching each of these we have 8 circles, forming ^8.7.6.5, =70, 
tetrads of circles: taking at random any one of these tetrads for the blue circles, 
the remaining red circle has to be determined so as to touch each of the four 
blue circles, that is, by four instead of three conditions; and there is not in general 
any red circle satisfying these four conditions. But the 8 tangent circles do not 
stand to each other in a relation of symmetry, but form in fact four pairs of circles; 
and it is possible out of the 70 tetrads to select (and that in 6 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 
c. xiii. 54