204 



DE THOMAS MUIR ON 



two top-to-bottom lines. The nine small square arrays thus resulting may then be 

 symbolized by 



• r -P 



-y ■ a 



fi -a 



each of the three-line matrices a, /?, y being zero-axial skew, and composed of 

 elements from only one row of the original determinant. 

 Now if we multiply this b}' | a 1 b 2 c s | 3 in the form 



h 



a 2 

 b, 



Co 



a. 2 

 b„ 



a. 2 



a, 



we obtain 



-B 2 

 -B„ 



B 2 

 B 3 

 0, 



c. 



"A, 

 -A., 



-A 3 



"A 2 

 -A, 



A x 

 A 2 

 A, 



C x -B, 



C 2 — B 2 



Cn — Bq 



-c. 



-o 1 



-c 2 

 -a 



3 



h | 

 Co I 



A x 

 A 2 

 A 3 

 B, 

 B 2 

 B Q 



which if divided like the multiplicand into nine three-line minors takes the form 



7 P 



-y 



Br 



-»i 



B 2 - 



-B, 



B 3 - 



"B 3 



<V 



-Ci 



C 2 - 



-c 2 



c 3 - 



-c 3 , 



-fi -a • . 



It is better however to write the three zero minors differently, viz.- 



Ax-A, . . 

 A 2 -A 2 . . 

 A 3 _ A 3 . . , 



for then we are able to state more easily the law of formation of the nine. Thus, we 

 may then say that the three 



• 7 /?' 



have -Aj, — A 2 , -A 3 ; — B v -B 2 , -B 3 ; — C l5 -C 2 , — C 3 respectively in their first 

 columns, and A v A 2 , A 3 in a column of each, the particular column being the first of the 

 first, the second of the second, and the third of the third. Similarly, the next three 



-7 



