( 102 ) 



If the above equations are to be satisfied then in every equation 

 at least one of the coefficients must become negative. 

 This is most profitably attained for n = 'It if 



«12 — ^m^i2<C^' "m — '^^'"''^a4<C^' '^^^^ fl-2t--],2t — ^mi^2l-\,2t<C^^ 



S0 then we have e.g. 



Us — h. &s > ^ <-ls,s-\-\ — ^-m l>s, .v+l <C <J "s+l — >*•/« '''5+1 ^ ^ 



SO 



tts s-\-l «^ «s+1 



-^ <; X,n and X,n < - and ;.„, < — — 



Os,s+l OS 6s+i 



SO that a fortioi'i 



<^ and <^ — 



The mixture [.v, .v -j- 1] then possesses a minimum. 



So at least minima are wanted in t binary mixtui-cs if I he wliole 

 mixture of 2t components is to show a minimum. 



If n=^2t-{-l then there are at least t-{-l or h{7i-\-'l) minima 

 wanted in the binary mixtures if the total mixture is to show a 

 minimum. Let us take e.g. 



ttj.^ — X,n /'i-^ <C ^^ ''34 — ^'"' '''34 <C ^ • • • • <^«— 2,H— 1 — hi K—2,u-l <C 



and 



«;il — A,„ b„i <[ 0. 



For // even the case is (aken that each conipoiiciit shows with 

 bill one other one a niinimuni, whilst for n odd c/h' of the compo- 

 nents gives a mininiiiiii with two other ones. 



If a component forms with more other ones miriima then more 

 conditions are al)solutely necessary; if a.o. we assume that// — lof 

 the components give mutually no minima, then certaiidy the last 

 component must give a uiiiiimuiu with each of the (/^ — 1) remaining 

 ones, if the total mixture is to show an absolute mininnim. 



Of course the above theorems may not be rcA'ersed ; so at least 

 ^ H. (resp. 4 {n -j- 1)) minima are wanted, but these do not in the 

 least guaranty the existence of an absolute mininimn in the //-foUl 

 mixture. 



In case of an attainable absolute maximum a,, — Aj/i^^.s is <^Oand 

 so in our set of equations at least one positive term is required to 

 present itself in each member. From 



a, — Xm 6s < 0, (Irs — /J/ 6/,s> and «,. — l^^^ b, < 



follows 



— >^jy, Avhilst /,i/>— and /j/>— , 



bjs bs Or 



