130 Solutions to Important Problems. 



the angle3 A, C, B which LM, MN and N O subtend at P are 

 known. 



The fundamental principles of " Modern Geometry " shew that 

 the anharmonic ratio of four straight lines radiating from any 

 point P is equal to the anharmonic ratio of the segments wbicn 

 they make on any straight line cutting them; and this theorem 

 gives us the equation. 



x (x+a + b) sin C. sin (A + B + C) 



r = ~j— 7 — tt~t3 — . a known number, 



a o sin A . sni t> ' ' 



which we may denote by h . 



x (x + a + b) 



And from the equation ; ■ = Ti, we at onceobtain 



a o 



— (a + b)+ J(a + 6)+4 a b h\ 



Now in this equation it is evident the radical must be taken 

 with the sign + : for if we suppose x to decrease until it 

 becomes = o, then sin C = o, and h = o, and we have 



i 



— («+&)+{ (« + &)} 



*= g"- =0; 



which would not bs the case if we were to take the minus sign 

 before the radical. 



Moreover it may be observed that if we find such that 



4<ab. sinC. sin (A + B + C) 



tan 2 = 2 



(a + b) . sin A . sin B 



2 2 



then D / , '- = h ; and the value of x takes the form 



4 a o 



—(a + b) + O + S) jl + tan 2 ys} 



x = ! I 



2 



(a + b) (sec 0—1) 

 2 



