OF SECONDARY MINORS OF A PERSYMMETRIC DETERMINANT. 



527 



We are thus brought to the following interesting identity : — 



X l 02 a 4 & 7o 



x \ 02 H Pb K 



P<9 



f 

 + ' *l 02 a 3 Pi C 6 



p,q; 3,4,5,6 



*i v% a s #i & 



(_l)p+»-l. 



In this, by reason of the mere presence of a Kronecker-Muir * aggregate, 

 ■p,q'- 3,4,5,6 I, as a factor in each term on the right, is included the fact that 

 the left-hand member vanishes when | a x fi 2 7 3 ^4 e 5 ^6 I * s axisymmetric. 



(24) Had the square array in the preceding paragraph been of the 5 th order we 

 should have had 



*1 03 «4 Pb 



9=2 5 



/ , I X P 09 

 P<9 



X \ 02 a 4 Jb 



+ 



p,q; 2,3,4,5 



X l 02 a 3 8 b 



X l 02 a 3 e 4 



but then the restrictions in the values of p and q being such that at least one of 

 the two must be the same as one of the variable group 2,3,4,5, the aggregate 

 p ,q ; 2,3,4,5 ! reduces to one of the next lower order. 

 Similarly when the square array is of the 4 th order we have 



\ X 2 03 a 4 - 



X l 03 Pi 



+ 



X l 02 74 



- 



X l 02 S 3 



9=2 , 3,4 



/| 1 X P 9 1 * 

 P<Q 



p,g; i, 



2,3,4 



•1 (-ir 



9-1 . 



) 





but as in every case p is one of the integers 1,2,3,4, and q another, the aggregate 

 is reducible in its order to the extent of two steps, thus becoming simply the difference 

 between a minor and its conjugate, as we have already seen (end of § 17). 



(25) Taking next an instance where the non-quadrate array has three columns, the 

 square array being of the 8 th order say, we may show exactly as in § 21 that 



X l 02 Z 3 a b As 77 S 8 



I X p 09 Z r I # 



p<q<r 



X l 02 Z 3 a i Ai 77 e 8 

 | X \ 02 Z 3 a 4 Pb 76 VS 



p,q,r; 4,5,6,7, 



X l 02 Z 3 a 4 Pb 77 ^8 

 + I X l 02 Z 3 a 4 Pb 76 °1 



(-iy + " +r , 



* So called perforce, as the less general expressions e.g. ' first brought to light by Kronecker 



already bear his name. 



TRANS. ROY. SOC. EDIN., VOL. XL. PART III. (NO. 22). 4 k 



