1905-6.] The r-line Minors of the Square of a Determinant. 535 
and so on. Therefore, just as in the previous case, the sum may 
be expressed in the very contracted form 
j 
rowj 
+ 
rowi + 
row 4 
+ r0W 2 
+ 
row 2 
+ r0Ws 1 
row 2 
row 3 
row 4 
lrow 3 
row 4 
row 4 J 
1 row r I 2 
+ 2^! 
row r 
row r , 
1 row, | 
row, ! 
row s , 
or 
where r, s is any pair of the integers 1, 2, 3, 4, and r\ s any other 
pair. Falling back, however, on the lengthier form, it is next 
seen that each of the 36 parts of it is expressible as the sum of 
six products ; for example, the first 
j* ^ a* j* f = l a AI 2 + l a A! 2 + l«AI 2 + IVsl 2 + i“AI 2 + \ a s b t? ■ 
so that altogether we have 216 products of pairs of two-line 
minors of the original determinant. Keeping an eye on those 
having only the suffixes 1, 2 we find under the first 2 
| afb 2 p + | a 4 c 2 1 2 + | aft 2 1 2 + | \c 2 | 2 + 1 h x d 2 | 2 + | c^d 2 | 2 , 
and under the second % 
2{|a 1 5 2 !-|a 1 c 2 | + I^AI'K^I + 
and therefore in all 
{! <h h 2 I + I «1 C 2 I + i <h d 2 I + I I + I h l d 2 I + I C l d 2 l} 2 * 
A similar result is of course got by considering any other pair of 
suffixes ; and, as there are six such pairs, our final result is 
{| ®A | + 1 1 ■+... + 1 c x d 2 1} 2 + '{| a A | + 1 «ic 8 1 + . . . + 1 cfL z |} 2 
+ {! «A I + I a \ C \ I + • • • + I C l d 4 I} 2 + {| a 2^3l + I a 2 C 3 I + • • ' + I C 2^3 I} 2 
+ { |a 2 & 4 1 + I a 2 c 4 1 + . . . + | c/Z 4 1} 2 + {| a 3 5 4 1 + | a 3 c 4 1 + . . . + 1 c 3 ^ 4 |} 2 . 
We have thus a theorem exactly analogous to that formulated 
in § 1 and reached in a perfectly similar way, — a way, too, which 
is seen to be just as readily applicable in the case of three-line 
minors, four-line minors, etc. The following generalisation may 
consequently be viewed as established — The sum of the r-line 
minors of A 2 is equal to the sum of squares , each of which is the 
square of the sum of the r-line minors formahle from a set of r 
columns of A. 
