( 97 ) 



stationary ; the A of that mixliire satisfies either the inequalities {A) 

 01' the ine(|nalities (B). 



Het'oi-c |)assiii<^ lo (he invcsligatioii ol" liie nature ot the stationary 

 point we shall first prove the theorem 



/;=1 7=1 



This theorem can he easily verilied tni- // =r 2 ; it holds sljll jxood 

 when bi,,j ^- l>,ip, i.e. for asymmetrie detei-minants. 



Tiie general proof is su[)plied by showing tiiat if llie theorem is 

 correct for determinaiils of [n — 1) rows and columns (also asym- 

 metric ones), it also holds good for d(Mcrminants of y/ rows and cohimns. 



Now we ha\e 



Let us now make use' of the following notation: 

 L,f„j is the determinant derived from L,^ by omitting the p^^^ row 

 and the ry^'' column. 



ti^jfj is the determinant derived from L^ by omitting the rows 



r s 



with numbers /; and r and the columns with numbers q and s. 

 We now find 



' -.-^ = - A, M,-b,, JA, ... - Inn Mm + V {au-^J'u) -^ . 



*=i 



Further we find 



J/,, = (-1)^-1 A,,. 



L\s is an (asymmetric) determinant with {ti — J) rows and columns, 

 for which we have supposed the theorem to hold good, so that 



-^-^' =\" -bu-Mir ± y (-l)-i (au— /-/m.s.) X 



d;i ^h^^ ^^w 



p^zn 7=:l,2,..(s) .71 



where the positive sign must be used for 7>.s' and the negative 

 one for (/ <^ s. 



Performing the summation according to .v first, we find 



Ï) By placing s belwocn brackel-s wc indicalc that for the summation all values 

 from 1 lo n except s must be assigned to q. 



7 

 Proceedings Royal Acad. Amsterdam. Vol. Vll. 



