( 101 ) 



Suppose an attainable mininiiini presents itself in llie mixture of 

 some components say 1,2,... />. 



Wiliiin (lie limits of possihility for llial mixtui-e a second set of 



values of llie ./'s foi- which L,, {) cannot be found. As farther --- 



t);. 



is negative for thai ininimiim we can dra\v Hie followinii,- conclnsions 

 from the system of ineijuaiilies (7'j): 



An attainable absolule mininnim bes lower than for the eom- 



h 



ponents and lower lluui eventnally appearing minima in any mixture 



to be compounded of the given components. 



If the original niixlure has a maximum and if there is also a 



maximum hi the />-fold mixture (J, 2, ../>), then for the maximum 



in the ^y-fold mixture L^, is equal (o and '' A^^i negative, so 



dA. 



—^ has there the sign of — A^,„.i. Now {—1)1'-"- A^,_i is < 0, so 



A/,_i has the sign of ( — l)p-^\ So y has the sign of ( — ly. Let 



-?y represent thai maxinunn, Ihen as A^, becomes but once for 

 possible mixtures A^, is furnished for every value of / ^ .^^„ with the 

 sign ( — 1)1', but for every value of ;.< P.^ with the sign ( — l)/'-i. For 

 the maximum in the /i-fold mixture ( — 1)"^ ' Lj, i.s <;[ 0, and so for 

 L^, the sign is indicated by ( — \)p. From this ensues that the set of 

 inequalities {1\) can be ex[>resse(l as follo\vs: 



An attainal)le absolute maximum lies highei- than the for the 



h 

 components and also higher than e\entual maxima in mixtures to 

 be formed of the given components. 



The question now arises whether a maximum or minimum in the 

 y^-fold mixture im[)lies anything about maxima or minima of the 

 binary mixtures to be formed of the y/-fold mixture. 



Su[)pose (he //-fold mixture to show a minimum for A = ;.„, and 

 the constitution of thai mixture to be indicalcd by b-rj,,,, (,/■ J,,,., .(,(;,,),„], 

 then w^e lind 



(a, — ).,nb^ j(''"i)m + (il,2 — hi l>xi) {■>-i),n +••••+ (♦«!« — K^\n){-l'n)m = 



(a.,1 — hn />,,) (''■ i)m + (". — hn />., ) ('i^'X + + (".,« — In h,„) {.V,,),, = 



[ (a„i — A;„ hu,) (■'', ),n + K, — hi Ih,^) {'^\)m -f + ("« — hn ''n ) {.>'n)m — 0. 



Now w^e know that for a possible absolute minimum //^ — l,,^ f>s^^ 

 whilst of course also {Xs)m > 0. 



