I04 KANSAS UNIVERSITY QUARTERLY. 



The condition for the tangency of the circle (12) and of the ori- 

 ginal circle (3) is 



Substituting the values of a, /3, p and a", /3", p'' and developing 

 we find the expression 



a^xb^ — 2aA 

 A"= — , 



2A(kdzl i + k2)— 2(ak4-b) 



which shows that to eacli value of A belong two values of A'', or 

 that each circle of the pencil 



x"^+y-— 2Ax=o 



is touched by two and onl_y two circles out of the pencil 



(x— a)2-i-(y_b)2_2A"(xk--y— ak— b)=o. 



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

 circles and it is for the present purpose not necessary to develope 

 further details. 



We will now show that any circle C of the pencil 



(x— a)2^(y— b)2_2A'(x— yk— a + bk)^o 



which is normal to a certain circle C of the pencil 



x2^ y2— 2Ax^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 

 Cg" out of the normal pencil of circles 



(x— a)2-u(y— b)2— 2A"(xk+y— ak— b)=o. 



In fig. 2, Cj" and C," are the two circles tangent to the circle C. 



Now tlie tangent to C or Cg" at B, intersects the tangent V^ in 

 the point (a'. /3', ), or N, such that NBr=NM = NA. Hence the 

 normal circle of C, C^", and Co" pass through A and B, q. e. d. 



The locus of the points of tangenc}' of the circles of our special 

 pencils of circles is, therefore, the same as the product of projec- 



