56 -P^'o/! Gayley, On a system of two tetrads [Feb. 13, 



them orthotomic to H, and touching the circles 1 and 2 : but a 

 circle orthotomic to Cl is its own inverse or image in regard to H ; 

 and it will thus touch the circle 3 which is the image of 2 in 

 regard to XI. 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 O, 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. 



5. The blue circles have a common orthotomic circle II, that 

 is the radical axis or common chord of each two of the blue circles 

 passes through one and the same point, the centre of the circle XI. 

 The figure is symmetrical in regard to the red and blue circles 

 respectively, and thus the red circles have a common orthotomic 

 circle XI', that is the radical axis or common chord of each two of 

 the red circles passes through one and the same point, the centre 

 of the circle Xl'. 



6. Projecting stereographically on a spherical surface, the 

 four red circles and the four blue circles become circles of the 

 sphere ; and then making the general homographic transformation 

 they become plane sections of a quadric surface ; we have thus the 

 theorem that on a given quadric surface it is possible to fiud four 

 red sections and four blue sections such that each blue section 

 touches each red section ; and moreover the capacity of the system 

 is = 9 ; viz. 3 of the red sections may be assumed at pleasure. But 

 (as is well known) the theory of the tangency of plane sections of 

 a quadric surface is far more simple than that of the tangency of 

 circles : the condition in order that two sections may touch each 

 other is simply the condition that the line of intersection of the 

 two planes shall touch the quadric surface. And we construct as 

 follows the sections touching each of three given sections : say the 

 given sections are 1, 2, 3 ; through the sections 1 and 2 we have 

 two quadric cones having for vertices say the points d^^ and \^ 

 (direct and inverse centres of the two sections): similarly through 

 the sections 1 and 3 we have two quadric cones vertices d^^ and i^^ 

 respectively, and through the sections 2 and 3 we have two 

 quadric cones vertices d^^ and i^^ respectively; the points d^,^, i^^, 

 ^13' *i3' ^23' ha ^^® three and three in four intersecting lines or 

 axes, viz. these are d^^d^^d^.^, ^^ishih^' ^mhs-hs' ^la^s^'si respectively. 

 Through any one of these axes say d^^d^^d^^, we may draw to the 



