226 



DR THOMAS MUIR ON THE 



and in the latter 



a y r 







*> f 2 



or — 



12 5 



13 4 



n I k 







But there exists the identity 



135 

 124 



125 

 134 



145 

 123 



= 



(K') 



consequently the cofactor of OX is 



145 

 123 



or 



123 



145 



as desired for the sake of symmetry in the square array of D . 



The identity (K') is noteworthy in that the terms of it are third-order minors of an 

 axisymmetric determinant of the fifth order, and not of the sixth order as is the case 

 in Kronecker's theorem. The best mode of establishing it is to make it dependent 

 upon the latter. Thus, taking the axisymmetric determinant of the fourth order 



2345 

 2345 



rs = sr 



we have from Kronecker 



23 

 45 



24 

 35 



+ 



25 

 3 4 



= 



(K) 



and this is simply what (K') becomes if 1 be struck out of each term of it. In other 



words (K') is the Extensional # of (K). The idea of applying the Law of Extensible 

 Minors to all Kronecker's identities is thus suggested. For example, knowing from 

 Kronecker that 



123 



456 



124 

 356 



+ 



125 

 34 6 



126 

 345 



= 



if the terms be minors of the axisymmetric determinant 



12 3 4 5 6 

 12 3 4 5 6 



rs = sr 



we can at once vouch for the identity 



12378 

 45678 



12478 

 3 5 6 7 8 



+ 



12578 

 3 4 6 7 8 



12678 

 3 4 5 7 8 



= 



in connection with the axisymmetric determinant 



12345678 



12345678 



rs = sr 



* Mum's Theory of Determinants, p. 213, § 179 ; or Trans. Roy. Soc. Edin., xxx. p, 2. 



