40 The Three Sections, Tangencies, 



C and D being the i^oints in which AI and BI cut MM and 

 NN, ive shall have PC to RD in a given ratio ofra to n." 



It is evident that tlie points G and O are coincident, and 

 .-. the solution of this enunciated problem can be worded as 

 folloAvs ; — 



Take Q and S on MM and NN, so that PQ : RS : : 

 m : n ; through A and B describe the circle AIB, which is 

 such that I being any point in it, we have the angle lA right 

 to B = ^ right; through P and Q draw PGr and QG, mak- 

 ing the angles PG and QG right to M, equals respectively to 

 the angles RB and SB right to N, through L where RB 

 again cuts circle ABI, draw AL to cut PG in H ; describe 

 the circle AHG ; through either point C in which it cuts 

 MM draw CA to cut circle A IB again in I ; draw IB to cut 

 NN in D ; then will PC : RD : : PQ : RS : : m : n. 



2. If we suppose not only I = k, but also = zero, and B 

 coincident with A ; then it is evident the problem becomes 

 the " Section of Ratio " of Apollonius. Moreover, it is 

 evident the preceding solution to the Apollonian problem 

 flows directly from the present more general problem, for in 

 this particular state of the data we evidently have G coinci- 

 dent with O, and B, L, and I coincident with A, and ARH 

 in straight line, &c. 



3. Since PC.SD : QC.RD : : I : k, if we suppose in NN 

 SR and RU always equal / and /c, we have PC.SD : QC.RD 

 : : SR : RU. 



Now, if we suppose R and U to remain fixed, and that S 

 becomes infinitely distant, then for a point D at a finite 

 distance, we have SR = SD, and therefore PC.RU == 

 QC.RD. 



Hence (QC— QP). RU = QC.RD, and (RU— RD) QC = 

 QP.RU, .-. DU.QC = QP.RU. 



Or, UD.QC = QP.UR = a known magnitude. 



Hence, we derive a method of solving the problem. 

 " Giiwn the j^oints U and Q in given straight lines NN and 

 MM ; through tivo given points A and B to draw two straight 

 lines AI and BI making the angle lA right to B equal a given 

 angular magnitude 9 right, and such that C and D being the 

 respective points in which AI and BI cut MM and NN, ive 

 shall have UD.QC = ni.n." Where m.» is given in sign, and 

 the directions on the given lines particularised. 



It is evident that in this case G coincides with P, and that 

 circle GPQ touches PG at this double point, and .•. that the 

 triangle QOP is similar to URB, &c. 



