( 100 ) 



Suppose the stationary point to be an absolute maximnm ormini- 



a 

 mum. Let X„ he the eorresponding value of -. Then for ;. = ;.„ there 



is only one constitution; thus for .c^, .v^, . . . ,€„ only one set of values 

 mav be found. Now Ihis is only possible when all tiie eoefticients of 

 Lj, L.^, . . . /v„-i iiave Ihe same sign for P. = ;.„. For, if this were not tiie 

 case then it would be possible for ;.=■ A„ (for which the last eoeflicient 

 has already (Usappeared) to satisfy tiie equation / =: wilhoiil the 

 necessity of L^, />.,, . . . Ln—\ l)eing individually zero and tiien many 



sets of adjacent values .c„.i\„ . . . .i',, might be found for whicli— 



had that absolutely maximal or minimal supposed \alue ; \\ hich is 

 absurd. 



For a stationary point to be an absolnte miuimiiui or maximum 

 it is therefore required 

 A,>0 ,A,>0 L,<C^) ,^i<0 



I , so ' :■ or i , S()(yj{ = 



Let ;.„ be a root of A„ = iudicating an attainable absolute 

 maximum or minimum, tiien for ).=:)•„ Hie coefficient of the last 

 square (.f„') in the development of / becomes zero. For / = ;.„ -\- e 



the sign of A„_i -— determines the sign of tluit coefticient, whilst 



dA„ 

 for A 1= ;.„ — f the sign is determined by — A„„i |. . 



Now however, as we just before ijuUcated, we lind, for a possible 

 stationary point 



So it is evident that for ;. = ;.„-f-s the last term is always negative 

 and for ^ = -^o — ^ always positive. 



From this ensues that in the case of an absolute minimum the 

 inequalities (7\) must be satisfied, whilst for an absolute maxinuim 

 the inequalities (7'J must be fulfilled. In the first case the conditions of 

 possibility {A) are still to be added, in the second case the conditions (B). 



It is clear that by a different numbering of the conq^onents other 

 inequalities would have been obtained — e\idently however the 

 system (Tj or (7\j is sufticieni to iiulicate an absolute minimum or 

 maximum. 



