38 UNIVERSITY OF COLOKADO STUDIES 



where the vij/s satisfy the linear equation 



All circles of the congruence are orthogonal to the fixed circle with 

 the equation 



Xj «,4-'*2 ^2 + ^3 % + ^4 a4=-0. 



Thus in corroboration of the previous result we may state the 

 theorem : 



A linear congruence of circles in a plane consists of all cir- 

 cles orthogonal to a fixed circle in the same plane. 



The properties of this congruence may be studied in a simple 

 manner by considering with M. Bocher, loc. cit., the sphere 



4 2 



2 a; =0 



1 k 



and its stereographic projection. Any cone of the second order with 

 the vertex V^ intersects the sphere in a sphero-quartic C ^. It is 

 well known that there are three other cones of the second order with 

 the vertices Fg, Fg, F^ which pass through the same sphero-quartic. 

 The tetrahedron Fj , Fg , F3 , F^ is the self -polar tetrahedron of the pen - 

 cil of quadrics passing through C ^. Now the polar -reciprocal surfaces, 

 with respect to the sphere, of the 4 cones are four conies K^ in the 

 planes of the polar tetrahedron. The planes of this tetrahedron 

 intersect the sphere in four mutually orthogonal circles. The tan- 

 gent planes of each cone intersect the sphere in circles which touch 

 C ^ in every case in two points, and whose poles are situated on the 

 reciprocal surface of the cone — in this case on a conic. Making a 

 stereographic projection of this configuration with respect to the 

 original sphere, the theorem may be deduced: 



The circles of a linear congruence lohose centers are situated 

 on a conic, envelope a hi-circular curve of the fourth order. The 

 same curve is also produced ly three other systems of circles belong- 

 ing to three different congruences, and having their centers on three 

 different conies, respectively. The fundamental circles of these 

 four congruences are mutually orthogonal. 



