104 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



where it is to be noticed that 23, 31, 12 are the tangential distances of the circles 

 2 and 3, 3 and 1, 1 and 2 respectively ; viz., if «, /3, y are the radii taken 

 positively, then these are the direct tangential distances. By taking the radii 

 positively or negatively at pleasure, we obtain in all four equations — the tangential 

 distances being all direct as above, or else any one is direct, and the other two are 

 inverse ; we have thus the four pairs of tangent circles. 



The cone-spheres which pass through a given circle are the two spheres which 

 have their centres in the two anti-points of the given circle ; and it is easy to see 

 that the foregoing investigation gives the following (imaginary) construction of 

 the tangent circles; viz., given any three circles A, B, Cm the same plane, to 

 draw the tangent circles. Taking the anti-points of the three circles, then select- 

 ing any three anti-points (one for each circle) so as to form a triad, we have in 

 all four complementary pairs of triads. Through a triad, and through the com- 

 plementary triad draw two circles, these are situate symmetrically on opposite 

 sides of the plane ; and combining each anti-point of the first circle with the 

 symmetrically situated anti-point of the second circle, we have two pairs of points, 

 the points of each pair being symmetrically situate in regard to the plane, and 

 having therefore an anti-circle in this plane ; these two anti-circles are a pair of 

 tangent circles ; and the four pairs of complementary triads give in this manner 

 the four pairs of tangent circles. 



I return to the equations 



( x -sy+(y-sy+(z -s-y =o, 



(a - S) 2 + (a - Sy- + (a" - S") 2 = 0, 



(b - sy + (v - sy + (b" - sy = o, 



(c - £) 2 + (c' - tf) 2 + (c" - sy = o ; 



by eliminating (S, S\ S") from these equations we shall obtain the equation of 

 the pair of cone-spheres through the points (a, a', a"), (b, b', b"), (c, </, c"). Write 

 x — s,y — S',z — S" = X, F, Z, then we have 



X 2 + F 2 + Z* = , 

 and if, for shortness, we put 



a = (a - xy + (a' - y y + <y - zy , 



B = (b - o:y + (V - yy + {b" - zY , 



c = ( C - xy + ( C ' - y y + {c" - z y , 



then by means of the equation just obtained the other three equations become 



A + 2[(a - x)X + (a! - y) Y+ (a - z)Z] = , 



B + 2 [(6 -x)X+ (V -ij)Y+ (b" - z)Z] = , 

 C + 2[(c - x) X + (c - y) Y + (c" - z)Z] = . 



