72 Colorado College Studies. 



Now as 21 was taken at pleasure on the circumference, it 

 stands in no special relation to the particular circle whose 

 centre is c, hence we may define the delta of an arc <f to be the 

 ratio which the tangent drawn from the extremity of 2<p to any 

 circle of the interior system bears to the tangent to the same 

 circle drawn from the origin of amplitude. 



If P and 31 be any two points of the circumference, as in 

 Fig. 2, — the arcs AP and All being respectively designated as 

 2<p' and 2<p", and the corresponding arcs of the circle of the argu- 

 ment as 2u' and 2h", — then the tangents from these points to 

 any circle whatever of the interior system are in constant ratio, 

 to wit, (In u' : dn u". Two limiting cases are to be noted. First, 

 the tangents in the case of one of the circles will form one right 

 line, the chord PM, which is then divided, in the ratio just 

 named, by the point of contact of the interior circle. Second, 

 the circle to which the two tangents are drawn may vanish in 

 the point G; it then appears that the two distances PG and 

 MG are to each other in the same before-mentioned ratio. 

 Combining these two statements, the triangle PGM has its base 

 P J/ divided at the point Tin the ratio of the sides PG: MG; 

 hence it follows that a line joining G^ to T would bisect the 

 angle PGM. 



Two consequences follow immediately. The first is the 

 promised second solution of the problem: On a given line to 

 find, by a geometrical construction, the point at which it is tan- 

 gent to one of a given system of non-intersecting circles which 

 have a common radical axis. The solution is: Find, as in the 

 previous construction, a point G, which represents a vanishing 

 circle of the system ; connect this with the two points P and M 

 in which the given line meets any other circle of the system, 

 (for in the general statament o£ the problem, just given, the 

 circle of the amplitude has nothing to distinguish it from the 

 others;) the point in which the given line is met by the bisector 

 of the angle PGM is the required piint of contact. 



The second consequence is still another geometrical defin- 



