1904-5.] Three-line Determinants of a Six-by- Three Array. 365 
(2) The product of any two complementary determinants of a 
six-by-three array is expressible in six different ways as an 
aggregate of three similar products. 
Taking as an example the product | a Y b 2 c z H/i g 2 h 5 | he. 00', we 
have from a well-known theorem by interchanging /, g , h in. 
succession with a 
a 1^2 c 3 M I = I /l^2 C 3 I'l a l9dh I + I tUfPtfz |*| /i«2^3 I + I h x b 2 c 3 H fl92 a Z I V 
i.e. 00' = 88' + 22' + 55'. 
By interchanging /, g , h in succession with b and /, g , h in 
succession with c two similar identities are obtained, viz. 
00' = 99' + 33' + 66', 
00' = IT + 11' + 44', 
which, however, it is simpler to view as derivatives of the first by 
cyclical substitution. On altering the order of the factors in the 
given product the same procedure leads us to 
O'O = 8'8 + 6'6 + 11 , 
O'O = 9'9 + 4'4 + 2'2 , 
0'0 = 7'7 + 5'5 + 3'3 . 
It is clear (1) that what is here done with 00' can be done with 
any similar product ; (2) that each product on the right, by 
reason of the mode of obtaining it from the product on the left, 
will consist of factors that are complementary ; (3) that the 
theorem used will not give more than six expressions, because 
the interchanging of two letters with two, — which is the remaining 
possibility, — is the same in effect as interchanging one with one. 
(3) The nine products in the first triad of expressions for 00', 
.... are the same as the nine in the second triad , and further 
can be so arranged that a row-and-column interchange will produce - 
the latter triad , any five of the expressions thus giving the sixth. 
Thus in the case of 00' such an arrangement is 
88' + 22' + 55' 88' + 66' + 11' 
66' + 99' + 33' and 22' + 99' + 44' 
11' + 44’ + 77' 55'+ 33' + 77' 
