MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 65 



In like manner 



ft- tr- 

 im 



/S'^a' 



= {yz + xy)(T — xyzb-, 

 = {zx -]-yz)<T — xyz c". 



Divide in turn by x, y, z and add, 



%xy c^.'S, X y y- 



xy z 

 Eliminate the product -t y z from the three equations above, 



X y -\- X z 13- a y x -\- y z y- a z x -\- z y 



Or' h- s ' h- c- s c- 



(41) 

 (43) 



(43) 

 xyz 



a" (7 



a- s 



Divide by the product xyz, there result three independent equations in 

 the reciprocals of x, y, z which in connexion with 



a; + 2/ + 2 = 1, 



determine these numbers and also s. Thus 



,1 , a- 6- — s^ 1 or.c — s"^ 1 _ 



y 



r-c- — s' 1 



b^ X ^ ^ y ■ b^ z 



y- a- — s^ 1 ,7^6- — s- 1 , „ 1 

 T2 ^H -2 — +y- — = 



. 1 



y IT- 



(44) 



= 0. 



(45) 



Eliminating x, y, z there results for determining s the equation 



2a=a^ a-V- + p-a- — S-, aV + y-a-— «= 



a-h- + p-a- — s-, 2/S-b-, p^-c^-^y-b'—s-, 



a-C- + y^a^ — S-, li-C--\-y-b' — S", 2y=C=, 



In this s is a symmetrical function of a, b, c and a, j8, y, these numbers 

 being respectively interchangeable. The equation is apparently a cubic 

 in s- but not so in fact, the term independent of s is zero. To show this 

 observe that 



