( 9ft ) 



... r=>i f)—n >j=,i 



or 



a 

 For a possible mixture for wiiich - becomes statioiuirv all (|iian- 



tities }L„i iicixe the same sign as is jyroved. It is now evideiii tluit 



for such points, Nvliatexer the sion oi J/ nia\ l)e, -I/,,,* is alwavs 



0/ 



negative. With the lielp of tliis theorem we ran investigate the eon- 



ditions to (Hstiiignish an absolute maximnni or minimum. 



Let us iu)w write F^a — )Jj. 



This expression /- regarded as function of ./'j, ./-.^ . . . ,/•„ and ). is 

 zero for every given set of values ./-j, ,/'.^, . . . .t'„, if we assign to / tlie 



value of belonging to that c()n>tilution. If invci'sely we start from 

 b 



a tiixen ). (= P.,,) tiien the sujulioiis of the ecpiation f ~ d — z^, h =: (), 

 regarded as an equation in ^\,.^:.^,....v,^ furnish all the mixtui-es for 



which - possesses the given value P.„. If luoreoxer that value P.„ is 

 h 



an absolute maximum oi' minimuin. tlicii only a single set of jiossible 



values .t\, .t',, . . . .'■« may satisfy that e(juation. 



As /^ is a homogeneous (piadratic exjjression in the // ./"s we can 

 write it down as the sum of // squares. 



Let us again call L„ the determinant forming the tirst member of 

 the equation in 1 of ohUm- //. A,, i the determinant derived from the 

 former by the omission of the /i'^' row and the «^'' column, Ln—i 

 the determinant obtained by the omission of the last two rows and 

 columns, so that tiually L^zU^ — 7.h^. 



The transformation into a sum of scpiares is brought about in such 

 a manner that the tirst square contains all the terms with .i:\, the 

 second all the remaining terms with ,c.,, etc., until tinally -t*^, is 

 onr last square. In order to evade surds we must every time 

 multiply our function i)y <letinite coefiicients. It is now exident that 

 by executing the develojnnent, if we represent the successive linear 

 homogeneous expressions to be s(piared by />,, A,, . . . />;„ we have : 



