1891.] Systems of Circles and Conies. 263 



Hence, taking as a particular case the inscribed and escribed 

 circles of a plane or spherical triangle, the following result is 

 obtained : — If we describe circles touching three by three the 

 inscribed and escribed circles of a plane or spherical triangle, we 

 obtain four sets of four circles, exclusive of the sides of the 

 original triangle and its Hart-circle ; each set forms a Hart-group 

 and in addition we can obtain twenty -four groups of four circles by 

 taking two out of any one set and two out of any other such that 

 each group is touched by two circles besides the two circles they 

 have been constructed to touch in common. 



Any group whatever of four circles of the eight that touch 

 any three given circles satisfy the above relation of condition, 

 some singly, some doubly, and some triply; and by taking all 

 such groups of four, and describing circles touching them three 

 by three the following result is obtained: — Eight circles can be 

 described to touch three given circles; these eight form fifty-six 

 groups of three ; to touch any three we can describe a set of four 

 circles exclusive of the original three, and one which with them 

 forms a group of four circles touching four others ; and we can 

 form a thousand and eight groups of four circles, two out of one 

 set and two out of another, such that each group is touched by 

 two common circles besides the two they have been constructed 

 to touch in common. 



These theorems are then extended to co-vertical cones which 

 are either circular or have double contact with a given one, and by 

 projection to conies having double contact with a given one. 



In the last case one of the results obtained is: — If four 

 straight lines X, Y, P, Q are such that through the intersections 

 of X with P, X with Q, Y with P and Y with Q there can be 

 described a conic having double contact with a given one (/>, then 

 the four conies touching X, Y and P and the four touching X, Y 

 and Q, all having double contact with 0, can be arranged in eight 

 groups each consisting of two touching X, Y and P, and two 

 touching X, Y and Q, such that each group, besides touching X 

 and Y, touch in common two tangent conies having double contact 

 with <f> ; and similarly for conies touching P, Q and X, and P, Q 

 and Y respectively. 



The enunciation of the reciprocal theorem is obvious. 



Another result is as follows : — Four conies can be described 

 having double contact with a given one <j>, and touching three 

 given lines or passing through three given points ; to touch any 

 three of these four conies sixteen conies can be described having 

 double contact with <£, exclusive of the original lines or points 

 and four Hart-conics; there are thus four sets of sixteen conies. In 

 addition to the groups of four touched by four conies having double 

 contact with <f> that can be formed by taking four out of the 



