286 On idnalytical Geometry. 



angles^ and coiisequenlly independent of x, y and z. Let us make 

 ,r::=?/=:Ac=l5 then , V^—z"^ 



As— -^i 



\ z he 



It thus appears that A and B may be treated as functions solely of 

 the angle BAG. Accenting, to designate the other angles, and we 

 have z^ —X" -^-y^ -^2[\ —2K)xy y^ ,.. 



z-'^x^^y"—2{l-2B)xy\ ^^ 

 y^^=^x''-\-z^-\-2{l-2k')xz-> , . 

 y^^x"--\-z''-2{l-2W)xz\ ^^ 

 x^^=y'-^z^^-\-2{\-2k")yz\ r^s 

 x^-=y^-+z' -2{V-2B")yz\ ^^ 

 From the identity of the second members of (1), (2) and (3), re- 

 sult A4-B=:l 



A'4-B'=l 

 A"+B''=l 

 Adding the corresponding equations of (1) and (2), and dividing by 

 2a?, of (1) and (3) and dividing by 2y, and of (2) and (3) dividing 

 by 25", we get 



a; + (l-2A)?/4-(l-2A>=:0 (4) • 

 (l-2A)a;+?/ + (l-2A'')c = (5) 

 (1 -2A>+(l-2A'%+z=0 (6) 

 From these three equations, we easily derive the equation of condi- 

 tion (1 -2A)=^+(1 -2A')"'+(1 -2A'0'+1 -2(1 -2A) (1 -2A') 

 (1 - 2A'') = 0, between the angles of the triangle ABC. 



Resuming the values of A and B, (observing that the difFerence of 

 the squares of two numbers is equal to the product of their sum and 

 difference) and 



16AB: 



x^y^ 



{x+y-{-z) (x + y-z) {x - y-{-z) {-x+y+z) 



X y X y 



in the same manner 



20?='?/^ +2x'>z'' + 2y''z'' -{x^+y'+z') 



16A'B'= 



X"Z^ 



2x^y^ +2a;-2:3 J^2y-z^ - [x^ +y^ +2« ) 



AB z^ AB z^ AW y2 



and consequently, A^B'^p' A^'^.^ '"'' k^'^'^ ^^^ 



