C. VIII. 
68 
546] 
THEOREMS IN RELATION TO CERTAIN SIGN-SYMBOLS. 
537 
independent as to its lines, and also independent as to its columns.: I derive from 
this the square 
a 0/3 Oy 8 Oe 
viz. in the new square the line d is replaced by bd, the designation of the columns 
will be presently explained. This new square is, by what precedes, independent as to 
its lines; we have to show that it is also independent as to its columns. 
As regards the columns, any column is either unchanged or it is changed in its 
fourth place only, according as the sign in b is for that column + or —; that is, if we 
write 6 = +, the columns of the new square are (as above written down) a, 6/3, 6y, 8, 6e; 
+ 
+ 
+ 
and 6 is a product of all or some of the original columns a, /3, y, 8, e: but as regards 
/3, y, e it contains an even number (or it may be 0) of these factors; for otherwise 
the sign in the second line of 6 instead of being + would be —. But these are 
the very conditions that show that the columns a, 6/3, 6y, 8, 6e are independent. 
Hence starting from the square 
- 
4- 
+ 
4- 
4- 
*+* 
- 
+ 
-4* 
4- 
+ 
+ 
- 
+ 
4- 
4- 
4- 
4- 
- 
+ 
+ 
4- 
4- 
+ 
- 
which obviously is independent as to its lines and also as to its columns; and trans 
forming as above any number of times in succession, we obtain ultimately a square 
which has for its lines any system whatever of independent roots, and by what precedes 
each of the new squares is also independent as to its columns; that is, every square 
independent as to its lines is also independent as to its columns. Q.e.d.