144 UNIVERSITY OF COLORADO STUDIES 



a" = a + X" k, 

 P" = b + X", (27) 



and p" = \"Vi + k*. 



The condition for the tangency of the circle (27) and of the original 

 circle (10) is 



(a _ a "Y + (j3- (3"y = (p± p") 2 . 



Substituting the values of a, ft, p and a", /3", p" and solving for X' ' 

 we find 



a 2 X b 2 — 2a\ 



2X (Jfe ± l 1 + £ 2 ) — 2 (a& + 6) 



which shows that to each value of X. belong two values of X", or that 

 each circle of the pencil 



X 2 _j_ y2 — 2 \x — o 



touches two circles of the pencil 



(x — a) 2 + (y — b) 2 — 2 X" (xk + y — ak — 6) = o. 



These results are all well known from the theory of pencils of circles, 

 and it is for the present purpose not necessary to develop further details. 

 We will now show that any circle C of the pencil 



(x — a) 2 + (y — b) 2 — 2V (x — yk — a + bk) = o 

 which is normal to a certain circle C of the pencil 



x 2 + y 2 ■ — 2 \'x = o 



cuts the latter circle in two points, A and B, which are precisely the points 

 of tangency of the two possible tangent circles C" and C" out of the 

 normal pencil of circles 



(x — a) 2 + (y — b) 2 — 2 V {xk + y — ak — b) = o. 



In Fig. 3, C" and C 2 " are the two circles tangent to the circle C. 



Now the tangent to C or C 2 " at B intersects the tangent V x in the 

 point (a', /3',), or N, so that NB = NM = NA. Hence the normal 

 circle of C, C/', and C 2 " passes through A and B. 



The locus of the points of tangency of the circles of our special 

 pencils of circles is, therefore, the same as the product of projectivity 



