OF SECONDARY MINORS OF A PERSY MM tTRIC DETERMINANT. 



525 



12 3 



4 5 6 



13 2 1 

 4 5 6 1 



1 2 4 

 3 56 



1 3 4 



2 5 6 



1 2 5 



3 4 6 



1 3 5 



2 4 6 



12 6 



3 4 5 L 



1 3 6 



2 4 5 



= o, 

 = o, 



when 



123456 

 12 3 4 5 6 



is axisymmetric* Now this theorem was extended by me in 1897, the 



nature of the generalisation being made clear and at the same time easily rememberable 

 by introducing the notation ! ; of § 1, for then we pass from Kronecker's to the more 



general theorem by simply writing 



for 



It is the aggregates of this more 



general theorem which we are now called on to consider, viz., such aggregates as 



12 3 4 



5 6 7 8 



12 3 5 



4 6 7 8 



1 2 3 

 4 5 6 



+ 



1 2 



3 4 



12 4: 



35C 



12 3 6 

 4 5 7 8 



1 3 



2 4 



1 2 5 

 3 4 6 



12 3 7 

 4 5 6 8 



+ 



1 4 



2 3 



1 2 6 

 3 4 5 



12 3 8 

 45 6 7 



to which may be prefixed 

 and its conjugate. 



, viz., the difference between a primary minor 



(22) A convenient short notation for these functions is obtained by writing the 

 variable line-numbers immediately after the invariable, and putting a semicolon to 

 separate the two groups : for example, 



1; 2,3,4 



, : 1,2; 3,4,5,6 j , ; 1,2,3; 4,5,6,7,8 



The following are their fundamental properties : — 



(a) The functions are alternating with respect to any two of the variable line- 

 numbers, or any two of the invariable : and therefore vanish when two numbers of either 

 group are identical, 



(b) If one of the invariable line-numbers be the same as one of the variable, the 

 function is reducible to one of the next lower order. 



For example, when, y in 



; x,y 



w 



is equal to a we have by definition 



* As I have already pointed out elsewhere (Proc. Boy. Soc. Edin., xxiii. p. 147), it is not necessary for the truth 



■ 112 3 4 5 6 1 



of only one of these identities that the whole of be axisymmetric, but merely a minor of it. For 



example, the first of the two here given, which manifestly may be written 



12; 3 45 6 



3 4 5 6 



holds if 



3456 



3456 

 in an amended form 



be axisymmetric. This explains the insertion of the theorem of § 17, which is really Kkonecker's 



