OF SECONDARY MINORS OF A PERSYMMETRlC DETERMINANT. 523 



For example, when the two given arrays are 



x i Vi 









a i P\ yi 



K 



«a y-2 









a 2 & y> 



h 



x 3 v% 









a s Ps y.s 



s 3 



x i Vi 









a 4 & y4 



s 4> 



\ - \ x l 



Vz 



A 



+ 



1 «i2/ 2 y4 1 - 



1 * 



the aggregate in question is 



I a 2 2/ 3 a 4 j ■■ I x 1 y 3 ^ i I + I .^y^ I •■ I ajj^a 



To establish the general theorem it will suffice to show that in the said aggregate 

 the cofactor of the element p q differs only in sign from the cofactor of the element q p , 

 and that the element r r does not occur. Now the latter statement is manifestly true, 

 for the appending of the r th column of the determinant to the given non-quadrate array, 

 A say, is immediately followed by deleting from the latter the r th row, so that the 

 element r r is thus struck out. Secondly, the term in which the element p q occurs 

 is ( — lY'^p where it occupies the place (q — 1 , n— 1) and has for its cofactor 

 (— iy~\ — iy- 1 + n - 1 multiplied by the determinant of the array got by deleting the 

 p th and q th rows from A : similarly it is seen that the term in which the element q p 

 occurs is ( — l) 7_1 D g where it occupies the place (p , n — l) and has for cofactor 

 (— 1) 9_1 ( — iy+ n - 1 multiplied by the same determinant: thus the two cofactors differ 

 only in sign, as was to be proved. 



If A (p q denote the determinant of the array produced by deleting the p ih and q th 

 rows of A, we may write our result in the form 



r=n 



2 



irT), =2 A*.«-{Pi-&}(-ir**^ 



For example, the aggregate above instanced, viz., 



I Wt** I - I x \ ."3 Pi I + I •'•i.'Ajyt I - I »i2/2 8 3 I 



= I X 3>Ji I ( a >-Pl) ~ I X ->!/i I ( a 3"7l) + i X iV3 I ( a 4~ 8 l) 



+ I *iy 4 I (Aj-y-j) - ! *\Vs I (Pi~h) 

 + I -hv-i I (y.- 8 3 )- 



(18) A suitable notation for such aggregates is obtained by writing the column- 

 letters of the non-quadrate array without their row-suffixes ; following this up, after a 

 separation mark, by the principal term of the given determinant ; and enclosing the 

 whole in rectangular brackets. Thus, the particular instance just used would then be 



[xij ; Uj/^S,], 



from which any one of its terms is got by simply taking the letters on the left along 

 with one of the letters on the right and affixing; to these three the suffixes of the remain- 

 ing letters on the right. The instance immediately preceding this is 



[■r. ;a 1 fS. 2 y. i ], 



standing for 



I X 2 a 3 I - I x^ I + I X 1 y 2 I, 



and vanishing when | a-ifioyz I * s axisymmetric. 



