OF SECONDARY MINORS OF A PERSYMMETRIC DETERMINANT. 



513 



Doing this, and noting the two conditions in regard to P 6 , we have the identity 



a, a„ 



\ h h 



«1 



P 



c i 



d x 





P 



«2 



C 2 



^2 



Q 



\ 



h 



h 



+ 



\ 



Q 



&3 



&4 



«8 



C 2 



C 3 



e i 





c i 



«8 



C 3 



C 4 



°4 



d 2 



C 4 



d< 





d x 



« 4 



C 4 



d 4 



(^ ^ 2 C 4 d A 



— a result which, when found, is readily verified by examining the cofactors of 



a i » ^i ' c i ' i > 



Similarly in the case of the identity of the 5 th order, 



t?2 ^2 ^3 



Q 3 

 Q4 



d d% 



Go 6. j 





1 



«1&2 



C 3^4 



«5 i 



















*1 



«2 



P 5 



P 6 



p 7 





P 2 



P 5 



« 3 



p 8 



P 3 





h 



Qi 



h 



&4 



h 





*1 



*2 



Qi 



h 



*6 



+ 



c i 



Q 2 



C 3 



C 4 



C 5 



+ 



C l 



C 2 



Q 2 



C 4 



C 5 





*i 



Q 3 



d 8 



rf 4 



*s 





f?j 



^2 



Q 3 



C? 4 



d 5 





e i 



Q 4 



e 3 



e 4 



e o 





e i 



e 2 



Q 4 



e 4 



e 5 





P3 



l'e 



P 8 



«4 



P10 





P 4 



Pr 



P 9 



P 10 



«5 





h 



6 2 



*3 



Ql 



*5 





&1 



^2 



h 



*4 



Qi 



+ 



c i 



C 2 



C 3 



Q 2 



C 5 



+ 



C l 



C 2 



H 



C 4 



Q 2 





<*1 



^2 



d 3 



Q 3 



^5 





d, 



d 2 



d 3 



rf 4 



Q 3 





e i 



e 2 



e 3 



Q4 



% 





e i 



e 2 



e 3 



e 4 



Q 4 



by putting 



"4 > "7 ' "9 ! "lO > ^4 = e i ' e 2 ' e 3 > e 4 ' a 5 ' 

 "3 > "e ' ■*■ 8 » ^3 > ■"• 10 = **1 ' 2 » 3 ' a 4 ' ^5 ' 

 "2 > "5 » ^2 ' "« > "9 = C l ' C 2 ' a 3 > C 4 > C 5 > 



and noting the two conditions in regard to P a , P Q , P 



9 ' A 10 



*, 



Q 



b. 



*3 



K 



^5 



"3 



C 2 



f.> 



''4 



C 5 



"4 



^2 



C 4 



d 4 



tZ 5 



H 



e 2 



C 5 



^5 



6 5 



, we 



obtain 











P 



«2 



C 2 



rf 2 



e 2 





h 



Q 



&3 



64 



h 



+ 



c i 



«3 



C 3 



C 4 



C 5 





d, 



«4 



C 4 



<h 



rf 6 





e i 



«5 



C S 



*5 



e 5 



«j « 2 a 3 a 4 a 5 

 & 1 6 2 6 3 h 4 h 



C l C 2 C 3 C 4 C 5 



d x d 2 c A d A d b 



e i e 2 H d 5 e 5 



which, again, can be readily verified by examining the cofactors of a x , b^ , c 1 , c/ x , e x , P. 



The cofactors of e 5 , it is worth noting in passing, are the three determinants of the 



previous case. 



The general theorem thus reached may be formally enunciated as follows : — 



If the elements of the determinant \ a ln | be any whatever subject to the condition 



that the last coaxial minor of the (n — 2) th order is axisymmetric, it can be expressed 



as the sum oftivo others formed from it by putting A 1 in the one and „ i 2 in the 

 other in place of n 12 , and making the new complementary minors of a n , a 12 the 



same as in the original and the new complementary minors of a 21 , a 22 the conjugates 

 of those in the original. 



If in either or both of the determinants on the right rows were changed into 

 columns the enunciation would be similar. 



