138 UNIVERSITY OF COLORADO STUDIES 



cular points, the curve passes through A T , A 2 , A 3 , A 4 , and twice through 

 the circular points. (14) represents therefore a bicircular quartic. 

 For b = o, c = o, a = d, X' — X, and (14) reduces to 



U t U 4 - U 2 U 3 = o, 



which represents a bicircular cubic. For X = — 1 the corresponding 

 circles V \ — U 2 =o and U 3 — U 4 = o are the radical axes of the 

 pencils. To sum up, the theorem may be stated: 



Two pencils of circles in which each circle of one pencil is orthogonal 

 to a certain circle 0} the other pencil, and conversely, are projective and 

 their product is a bicircular quartic. 



If the radical axes of the two pencils are orthogonal, then the curve 

 degenerates in a bicircular cubic and the line at infinity. 



II. BICIRCULAR QUARTICS PRODUCED BY PAIRS OF TANGENT CIRCLES 

 OF TWO PENCILS OF CIRCLES 



In view of the application to compound curves, we have to show how 

 the foregoing bicircular quartic may be produced by tangent circles, 

 as indicated in the above title. 



Let W be a circle of the pencil through the two fixed points A 1} A 2 

 Fig. 1. Now, it is known that in general there are always two circles 

 through two other fixed points B l} B 2 tangent to W. To construct these 

 draw a trial circle K through B x and B 2 , cutting W in C and D, and let 

 CD produced cut B Y B 2 produced at N . From N draw the tangents 

 NA and NB to W: then A and B are the points of tangency of the 

 required circles V z and V 2 with W. AE and BE produced, cut the 

 bisector of B I B 2 at the centers G and F of V z and V 2 . If we now con- 

 struct the circle W ' , with iV as a center and NA = NB as a radius, then 

 W is orthogonal to W, V iy and V 2 ; W is therefore a circle of the pencil 

 through A 3 and A 4 conjugate to the pencil through B z , B 2 . In Fig. 1 

 the points A 3 and A 4 are imaginary, since B z and B 2 are real. The 

 same holds true for every circle of the pencil through A , and A 2 . Hence 

 the theorem: 



The locus 0} the points 0} tangency of pairs of tangent circles belonging 

 to two different pencils of circles is identical with the product of one pencil 



