526 



DR THOMAS MU1R ON VANISHING AGGREGATES 



a , b ; x , a , z , w j = 



which is simply 



or 



= 





a b x 

 a z to 





aba. 



; x z to \ 



+ 



a b z 

 x aw 



- 



» a z 



= 





a b x 



a z to 









- 



a b z 

 a x to 



+ 



a b to 

 a x z 





b x ; 



Z tO : 



b; 



b Z ! 



+ 

 X W : 



X,Z,W \, 



; b to 

 j x a 



; » 













_ 



__'• 











in regard to a determinant of the next lower order. 



(23) With these auxiliaries let us now return to the development of '£( — l r ~ 1 D r in 

 §18, beginning with the instance where the two arrays are 



x i V\ 



x i v-i 



x i y* 



x 5 y 5 



X G Vfi 



Ai 7i 8 i € i £i 



/3 2 y 2 8 2 Co £ 2 



Ai 73 S 3 e 3 t S 



Pi y 4 K H U 



Ai 75 8 5 ^ ^5 



A) 76 8 6 e 6 U, 



and where therefore the first agoreerate is 



a; i2/2 a 4/ 3 5 7(5 



S iy2 a sA 8 6 I + ! ^ll/i'hPi^ 



4 e 6 



^y^sl^iU 



Expanding each of the four determinants here in terms of minors of the first two- 

 columns and the complementaries of these minors, we obtain 



X l V-I I • { i a 4 A 5 76 I I a 3 Ao 8 6 I + I a 3 A4 £ 6 I I a 3 A 4 4 I : 



*l?/3 



+ 

 + 



+ 



{ i a 4&76 I 

 {I « 2 AA i 



a 3 As 8 6 

 a 2 A4 e <5 



a 3Al e 6 I 



^ 5 2/ti i * { i a iM>74 I - I «i/3 2 8 3 I }• 

 The cofactor of \ x 1 y 2 \ may also be written in the form 



1 2 3 



45 6 



1 2 4 

 3 5 6 



1 2 5 

 3 46 



1 2 6 

 34 5 



and therefore is equal to 

 Similarly we have 



1,2; 3,4,5,6 



cof 



x i v& 



1 3 4 



35 6 



1 35 

 3 4 6 



1 3 6 

 34 5 



and finally 



cof 



* 5 2/r, 



1,8; 3,4,5,6 j; 



: 3 5 6 : j 4 5 6 



|456: ~ |356 



5,6; 3,4,5,6 



