364 
Proceedings of Eoycd Society of Edinburgh. [sess. 
The Three-line Determinants of a Six-by-Three Array. 
By Thomas Muir, LL.D. 
(Second copy of MS. received September 12, 1904.* 
Read November 7, 1904.) 
(1) If the array in question be 
a i \ e i A 9t K 
b 2 c . 2 / 2 9 % ^2 
«S *3 C 3 fi 9s h s . 
its score of three-line determinants | af 2 c 3 | , | uf> 2 f z 
may be viewed as consisting of two complementary sets of ten, 
each of the first set containing at least two columns taken from 
| a 1 b 2 c z | , and each of the second set at least two columns taken 
from \f 1 g 2 h s I . Further, either set of ten may be viewed as 
consisting of one unique member and three sub-sets of three 
members each, the members of a sub-set being derivable from one 
another by performing the cyclical substitutions 
In this way a convenient notation for the twenty determinants 
will be found to be 
i a f^2 C 3 1 
0 
1 ^ 1 ^ 2/3 1 3 1 ^1 C 2^3 I > 1 C l ft 2^3 1 
1, 2, 
1 ^1^203 1 ’ 1 ^1 C 2^3 1 J 1 ^ 1 ^ 2/3 1 
4, 5, 
1 1 , 1 ^ 1 ^ 2/3 1 ’ 1 C l ft 2^3 1 
17, 8, 
1 fi9jh 1 
O' 
1 c i9fh 1 3 | ^ 1 ^ 2/3 1 ’ 1 ^ 1 / 2^3 1 
i', 2 ', 
1 ^ 1 ^ 2/3 l 3 I ^ 1 / 2^3 1 » 1 ^\9f l z 1 
* f - 
4' , 5' , 
1 2.^3 1 3 1 a i9f-z 1 s 1 bf 2 fo | ^ 
,7', 8', 
3 
6 
9 
3' 
6 ' 
9' 
* The original MS. was despatched by the author from Cape Town on 
20th March 1904, but was lost in transit through the post. — [Sec. R.S.E.] 
