136 UNIVERSITY OF COLORADO STUDIES 



For every value of X we get two conies which generally intersect each 

 other in four points. For all values of X the locus of these points is in 

 general a curve of the fourth order, whose equation 



(a U l — bU a )U 4 — (cU 1 —d U 2 ) U 3 =o (3) 



results from the elimination of X between (1) and (2). 



For the purpose of this paper we assume in the first place two pencils 

 of circles. The first of these shall be determined by two circles through 

 the points A, and A 2 , with the equations 



U l = x 2 + y 2 — 2a, x — 2b 1 y J r a, 2 + b t a — r t 2 = o, (4) 



U 2 = x 2 + y 2 — 2a 2 x — 2b 2 y + a 2 2 + b 2 2 — r 2 2 = o; (5) 



the second by two circles through the points A 3 and A 4 , with the equa- 

 tions 



U 3 = x 2 + y 2 — 2a 3 x — 2b 3 y -j- a 3 2 + b 3 2 — r 3 2 = o, (6) 



U A = x 2 + y 2 — 2a 4 x — 2b 4 y + a 2 + b 2 — r 2 = o. (7) 



Designating the parameters of the pencils of circles determined by 

 (4) and (5), and (6) and (7), by X and X' respectively, we have as equa- 

 tions of these pencils 



a. -\- Xa 6 + \b 2 ._. 



U 1 +\U 2 =x 2 +y 2 -2 \ +x x-2 J Y^~y (8) 



, a, 2 +Xa 2 2 + V +\ b 2 2 —r 1 2 —\r 2 2 



+ TFx = °' 



a, + X' a, b.-\-\'b. . x 



U 3 + X'U^x 2 +y 2 -2 3 i+x , 4 x- 2 \ +x , 4 y (9) 



a,' + X- a 2 + b 3 2 + X- b 2 -r 3 2 + X' r 4 ' _ 



+ i+X' 



For certain values of X and X' two corresponding circles of these 

 pencils are orthogonal, if the condition holds : 



