KANSAS UNIVERSITY QUARTERLY. 



The circles of the first pencil we assume tangent to the y — axis 

 (x=o), and those of the second tangent to the line 



X — a 



=k, 



y— b 

 or X — vk — a-[-L)k=o (2) 



at the point (a,b ). 



Now any two circles of a pencil of circles determine all the other 

 circles of the pencil, as indicated in the formulae (i). We may 

 also choose special circles of these pencils, for instance, the tang- 

 ents at their double points and the zero-circles in these points. 

 Thus we have 



' U^x2+y2, V=x 



U'=(x— a)3 + (y— b)2, V'=x^yl<— a + bk. 

 The equations of our special pencils of circles are therefore 



