530 



DR THOMAS MUIR ON VANISHING AGGREGATES 



referred to cannot be made without a simultaneous extension of the condition of axi- 

 symmetry. This accounts for the use of the word " apparent " in the italicised clause 

 near the beginning of § 26. 



For the same reason the notation for Kronecker aggregates given in § 18 may be 

 used to express the theorems of §§ 23, 24, 25. For example the first of these may be 

 written — 



[ x y a p ; a : /3 2 y 3 S 4 e 5 £ 6 ] = \ x x y 2 



+ 



[ a/3; y 8 8 4 c 6 £ 6 ] - 

 + 





where the occurrence of one or more letters on both sides of the semicolon in [ 

 results in the vanishing of one or more terms of the aggregate so denoted. 



(29) We can now return better equipped to consider the theorem of § 16, from which 

 we were led aside by the need for investigating certain auxiliary results. And, first of 

 all, the said theorem can be generalised as follows : — 



If two determinants of the n th order S , S' and an array X consisting of n — 2 

 columns ofn. elements each be taken, and a new determinant of the (2n—l) th order, A r > 

 be formed with zero elements in the places of its first coaxial minor of the (n — l) tA order, 

 with the conjugate of S' for the complementary minor, with the array X and the r th row 

 of S in the as yet unappropriated left-hand bottom space, and with the remaining 

 rows of S in the conjugate space, — then 



r=l 



is equal to 8 multiplied by an expression which vanishes when $' is axisymmetric. 



The mode of proof employed in § 16 applies generally and need not be rehearsed. 

 As for the vanishing cof actor of 3 it is nothing more than a Kronecker aggregate in the 

 form which appears in § 17. 



(30) If we take one of the minors formed from X. by the deletion of two rows, and 

 equate its cofactor on the one side of the identity of § 27 to its cofactor on the other 

 we have the following theorem : — 



If a determinant 8 of the n' ft order and an array y consisting ofn rows and two 

 columns be given, and from this a determinant A r of the (u+ \) th order be formed by 

 taking all the rows ofS except the r th , placing under them as rows the two columns of 

 y, and prefxing a column consisting of n — 1 zeros and the tivo last elements of the 



Jh 



roiv ofS, — then 



^AX-ir 1 



is equal to 8 multiplied by the difference between the (n— l) th element of the 2 nd 'column 

 of y and the n th element of the first column. 



