228 DR THOMAS MUIR ON THE 



which being a bordered zero-axial skew determinant is expressible in the form 



<T I V —/U T 



X -a 



where, be it observed, a- is the Greek letter in the place (4, 2). 

 We thus have 



cofactor of (1,3), i.e. g -m , in A 



= G - 



2 nd row of D 



(p,<r,\,T,/u,i / ) 



+ 



— O , — T 



. " 











p 





4 th row of D 







G 













X 



Similarly 









cofactor of (1,4), i.e. r + r , in A 









_ T? i Ond ^/% m *r r» / \ _ .. .A 



.^ 



X JUL V 





— XI T 



U J. \J VV Ul u 



v/-»,o-,/\,t-,^,i// 









3" row of D 







R 



P 













X 





+ 



cr\ v —/j. t 

 X -a- 



+ XI 



■V T 



X -a- 



A reference to the original determinant A shows that the cofactor of (3, 1) differs 

 from the cofactor of (1, 3) simply in having all the Greek letters of the opposite sign. 

 Consequently 



cofactor of (3,1), i.e. g + v , in A 



= G + 



T , V 



2 n " row of D 



(p,ar,\,T,fX,v) 



+ 









4 th row of D 







G 



P 



IT 

 X 



— <T\ V 



■ /J. T 



X -a- 



(13) The relation between the cofactor of (r, s) and the cofactor of (s, r) just referred 

 to shows that the adjugate of A is a determinant of the same form as A, and therefore has 

 a development similar to that of A as given in § 9. But the adjugate of A is equal to 

 the 3rd power of A ; consequently we may equate the one development to the 3rd 

 power of the other, — a process from which one or two curious identities arise. 



A similar result follows from the verification of the identity 



A - (l,l).cofactor of (1,1) + (l,2).cofactor of (1,2) + ... 



by substituting for A, cofactor of (1,1), cofactor of (1,2), . . . their developments 

 obtained in §§ 9, 12. 



Tims, recurring for shortness' sake to the 3rd order, we have the identity 



X Y Z x y z 



X a h (j 



Y h b f 

 Z 9 f ,: 



A 



H 



G 



H 



B 



F 



G 



F 



C 



X 



a h g 



V 



h b f 



z 



9 f o 



