700 
Transactioyis of the Royal Society of South Africa. 
Instead of all this the other procedure only requires that we write down 
the simply identity 
Ia3yl • 11231 = |a/3l| • |y23i + |n/32l • ily3| + I«/33l • |12yl, 
where the determinants are denoted by their column-numbers, and put the 
triad a, /3, y equal to each of the 20 triads 
1, 2, 3; 1, 2, 4; . . . ; 4, 5, 6, 
in succession. Line for line the outcome is the same until we come to the 
last, when we obtain at once 
■ |456| • ,123i = 11451 • 12361 - !245i 11361 + 13451 • 11261 
20-1 = 8-13 - 14-7 -r 17-4, 
which is what Yahlen's last reduces to when we express the compound 
determinant in it in terms of the elements of the last row and their 
complementary minors. 
(9) It is interesting to note in passing that the compound determinant 
just referred to has every one of its nine primary minors exactly divisible by 
lajfegCgl ; indeed Yahlen's first nine relations are statements to that effect. 
The expression for 20*1 which the first row and its complementary minors 
have just furnished can thus be matched by using any one of the other rows 
or any one of the columns, the six equivalents obtained being 
ll-lO - 516 + 219, 
-12-9 + 6-15— 3-18, 
13-8 - 7-14+ 4-17, 
11-10 -f- 12-9 + 13-8, 
- 5-16+ 6-15- 7-14, 
219 - 318 + 417. 
On this point may be consulted a paper in the Proceed. B. Soc. Edinhurgh, 
XXV (1904), pp. 366-371, on "The three-line determinants of a six-by-three 
array." 
(10) The property utilised in the preceding paragraph holds for all 
orders, and deserves to be noted quite apart from its connection with the 
present subject. Stated in its simplest form it is that every minor of a 
Bazin compound determinant is a determinant of the same kind. 
(11) The additional term for the expression 20*1 is, of course, due to the 
fact that the determinants 1 and 20 have no column in common. As, however, 
there are nine of the determinants of the array that have one column in 
common with 20, there are necessarily nine expressions of the shorter type 
which all involve 20, and of which one could be used to make the set of ten 
relations more uniform ; for example, 
20-2 = 8-15 - 9-14. 
