( 96 ) 



(-1) ^1A:>'J -^1A.>0 .1/1. > O 



(B) M,,<^0 M,,<0 .l/i„<0. 



We have just now made use of the last {u — 1) equations of 

 system ( V) to calculate the x's. As we might have chosen {n — 1) other 

 equations it is evident that for the existence of a stationary point 

 Af^,^ must always be either ^0 or <[ 0. The set of inequalities 

 given above is however sufficient to judge the possibility of the mixture. 



At most 07ie of these n mixtures can be realized. Suppose that two 

 different possible mixtures were to be found 



[(•^i)!, M,> ' ■ ■ M,] and [(.x'l),, (,u,)„ . . . (.i'„)J 



a 

 for which - were stationary. Now as in consequence of the set of 



equations ( V) equal roots P. lead to equal constitutions, different 

 values A belong to different mixtures. If we call the roots ?. belonging 

 to the above mentioned mixtures P., and Aj we arrive at the following 

 sets of equations 



da\ /"db^ ^ //öa\ fdb\ 



-' a- '■©="- ■-■(!),- '■©,-■ 



Multiplying the equations V^ resp. by {d\)^,{x„)^,...{.r„)., and those 

 of the system T\ resp. by ( — .v,)„ ( — .v^)„ . . . (— .v„)i '^"d summing 

 these up we find in connection with the identity holding good for 

 homogeneous quadratic functions 



v,.,,.@,|,„,(|). 



= 



All b^cj'a however being positive and all .vs also for possible mix- 

 tures, the above mentioned equation cannot be satisfied. 



a 

 So there is at most but one mixture possible for which - becomes 



