1904 - 5 .] Three-line Determinants of a Six-by- Three Array. 369 
(7) Turning now from the products whose factors are comple- 
mentary to those whose factors are not, we see that the taking 
of 0 along with any other of its own set (e.cj. 01, 02, . . .) would 
be nugatory, because the two factors of any such product would 
have two columns in common. But 01, 02, . . . , 09 being on 
this account unfruitful, it follows that the same cannot be said of 
01', 02', . . . , 09'. As for the products which begin with 1, they 
must be nine in number, because if they cannot be taken along 
with any particular one that follows it in its own set, this very 
fact ensures fruitfulness if taken along with the corresponding one 
of the other set : as a matter of fact the useful cases are 
12, 13, 14', 15, 16', 17', 18', 19. 
Similarly the useful products beginning with 2 are 
23, 24', 25', 26, 27, 28', 29' ; 
those beginning with 3, 
34, 35', 36', 37', 38, 39' : 
and so on. It is thus seen that if we confine ourselves to the 
products whose first factor at least is taken from the first set of 
ten and is represented by a smaller integer than the second factor, 
the number of fruitful products is 
9+8+7+ ... +3+2+1 . 
From every one of these products, however, another fruitful 
product is obtainable by changing each factor into its complemen- 
tary. The total number is thus 90. 
(8) Taking the first of the ninety, viz. 01', we have on inter- 
changing c, £/, h in succession with a 
I a f > 2 C B M c \9f l s j = i #1^2 c 3 H C ’\ a fH I + I hf>2^2 H C l92 a B I > 
i.e. 01' = 23 - 59. 
Now, no new result is got by interchanging c, g, h in succession 
with b, nor by interchanging c, g, h in succession with c. 
Further, by reversing the order of the factors in 01' and 
applying our theorem, we merely repeat the same result. We 
thus learn that each of the ninety products of pairs of non-com- 
plementary three-line minors formed from a six-by-three array can 
PROC. ROY. SOC. EDIN.— VOL. XX Y. 24 
