60 Colorado College Studies. 



In this construction there is evidently an ambiguity. We 

 may draw the tangent to the circle T^,/ST\, at either point F, or 

 Fo, and join its terminal points upon the given circle with either 

 Pi or Pg. It is evident, however, that we do not thus obtain four 

 results, but only two; since the two sides of the triangle P^MiNi 

 will, on the movement of P, envelope the same circle as those 

 of Poil/ajVo, and those of PiM.X. the same as PJI^Ni. 



The analytical parallel to the geometrical construction just 

 given consists in determining a and r, the elements of the circle 

 TT'W, in terms of those of Fj/SFo, that is, in terms of g and j9. 

 And the double solution of the geometrical problem answers to 

 the ambiguity in the sign of j), of which, in equation (9), we 

 fixed only the absolute value. AVe now notice that m' is the 

 abscissa of the point o£ contact of the base of that triangle 

 whose vertex is at Pj, where c = P; and m" corresponds in the 

 same way to the vertex at Pa, where c = — P. Hence if the 

 base of the triangle is on the same side of the circle FjSFo as 

 its vertex is, — as in the figure, — we obtain the radius of the 

 circle with its positive sign by the formula p = | (m' — m"), 

 but if the triangle were F^MoNo the radius would be of oppo- 

 site sense and denoted by the formula jj = ^ (m" — m'). As- 

 suming that the former is the case, we have 



p ~'r 9 — — *^" 9,nd j; — g = m' 

 whence, by equation (8), 



and 



If we take from one of these the value of r- and substitute it 

 in the other we obtain for the unknown quantity a a quadratic 

 equation, whose roots are 



« = flJ^+P ± V(P+i^)-'-r]. (12 



