534 Proceedings of Royal Society of Edinburgh. [sess. 
and the general theorem — The sum of the elements of the square of 
a determinant of the n th order is expressible as the sum of n squares , 
each of ivhich is the square of the sum of the elements of a column of 
the original determinant. 
(2) It is easily seen what change is necessary when, in squaring, 
the multiplication is performed in column-by-column fashion. 
When row-by-column multiplication is used, the result is no 
longer a sum of squares but is a sum of binary products each of 
which has for its first factor the sum of the elements of a row, 
and for its second factor the sum of the elements of a correspondiug 
column. Denoting by R r the sum of the elements of the r th row, 
and by C r the sum of the elements of the r th column, we see there- 
fore that the sum of the elements of A 2 
= Cf + C 2 2 + . . . -1- C n 2 when A 2 = A x rr A , 
= ~Rf + R 2 2 + . . . 4- R n 2 when A 2 = A x CC A , 
= R 1 C 1 + R 2 C 2 + . • . + R n C w when A 2 = A x rc A . 
(3) Turning now to the 36 two-line minors of A 2 we see that 
they are 
11 
12 
11 
13 
11 
14 
1 12 
13 
13 
14 
12 
22 
, 12 
23 
5 
12 
24 
, ' 22 
23 , , 
23 
24 
11 
12 
11 
13 
13 
14 
13 
23 
, 13 
33 , 
33 
34 
11 
12 
13 
14 
14 
24 
, . . . 
34 
44 
13 
23 
1 
33 
34 
14 
24 , . . . 
34 
44 
the array being of course axisymmetric. The first of the 36 is 
CL i $2 ^3 ^ 
a 1 a 2 a s a 4 
or 
row 2 ! 
\ \ h h 
\ b 2 b 3 b t 
row 2 
the second, which occurs twice, is 
rowj 
row 4 1 
row 2 
r0W 3 
