36 UNIVERSITY OF COLORADO STUDIES 



side of the plane and the circles in this plane. All circles of a plane 

 form a complex of circles. An analytic relation 



between the Cartesian co-ordinates a-, y, s oi a, point in space defines 

 a surface and limits the points in space to a twofold continuum. 

 The corresponding system of circles, consisting of a doubly infinite 

 number, is called a congruence of circles. Assuming the plane of 

 reference as the A''J^-plane, and writing the equation of the surface 

 in the form 



z=^ (a?, y), 



the equation of the congruence of circles is 



{^-xY + {v-yY=[f[x,y)Y, 



where f and 7; are the running co-ordinates of any point of the con- 

 gruence. The congruence is of the same order as that of J^' (;r, y, s) 

 = 0. Thus, in general, about every point of the plane of reference 

 there are n concentric circles of the congruence, n being its order. 

 As an example, take for F (a?, y, s)=0 the plane 



ax-\-hy-\-cs-\-d=0. 



The equation of the congruence becomes 



The equation of s is 



ax-\-hy-['d = 0, 



and the distance of the center (a*, y) from s is," Fig. 1. 



ax-]rhy-\-d 



A'B=- 



l/«2+52 



The radius r=A''L= — -., hence 



cos (b= y-^r^ =Fi 



