76 
Proceedings of Royal Society of Edinburgh. [sess. 
and thence eliminate a 2 , a v The result is 
tt ll^2 C 3^4l — ^41^1^2^31 J&jCgC^J l^l c 2^4l 
^ll ^2^3^51 1 rt 5l^l C 2^3l i^l C 3^5l l^l C 2^5l 
^ll^2 C 3^6l ” ^gI^1 C 2^3I h°A\ h^d Q \ 
M * * a 
i.e. 
i.e. 
0. 
\\0 z d,\ \\c,d,\ 
|^2 C 3^5l l^l C 3^5l |^l C 2^5l a h 
|^2 C 3^6l l^l C 3^6l l^l C 2^6l % 
= 0 , 
. 1 
d± Cj b 1 a x 
h c i\ -\ h A\ • 
d± c 4 & 4 a 4 
H^i^sl l c :AI * 
^5 C 5 ^5 a 5 
l^2 C 3l 1^2^31 l C 2^3l 
d 6 c 6 b 6 a 6 
= 0 
so that on dividing by \\c 2 df? we have 
{af^df- 0 , 
as was to be proved. 
Let us pass now from the determinants of the 4th order arising out 
of the rectangular array with which we commenced to the determinants 
of the same order arising out of the square array 
h h 
d} d : > d 2 d i 
h h 
d- 0 
C 1 c 2 °3 °4 °6 
/i /a fs ft A A 
and let us inquire how many of these minors are independent. We 
know that the total number of them is (C 6 , 4 ) 2 , i.e., 225, and that 
they constitute the elements of the 4th compound of \a 1 b 2 c s d 4 e 5 fQ\ . 
We see further that the first row of this compound determinant con- 
sists of the fifteen determinants dealt with above, and that therefore 
only three of the fifteen are independent. Similarly it follows that the 
vanishing of the first three of the second row entails the vanishing of 
all the rest of the row, and that the vanishing of the first three in the 
