Properties of Pfaffians. 
211 
As the like operation performed with the 1st and 3rd hnes of 
be 
C4 C5 Cg 
de 
^6 
leaves it unchanged in form, its value must be 0, as we have seen. 
7. Going still a stage further back we have the theorem : TJie value of 
a Pfajfian remains unaltered if the elements in the part of the rth line 
which is parallel to a part of the sth line be increased by m times the 
corresponding elements of the latter, the element common to the tivo lines 
be left unchanged, and the remaining elements of the rth line be 
dhninished by m times the remaining elemeyits of the sth line in order. 
For example, when the two lines are the 2nd and 5th we have 
1^2 
b, b. 
a^ ag 
b, be 
d. de 
a. 
a, 
ae 
-md^ 
^5 
be + mee 
ce 
d^ 
de 
and when the lines are the 5th and 2nd we have it equal to 
ae 
b. 
a^-j-ma^ 
C5 - mb^ 
d.—mb. 
ce 
de 
Ce + mbe 
The theorem, like its analogue, may be effectively used in making 
' evaluations.' Thus 
|2 
4 
5 
6 
= |2 
3 
4 
5 
1 
6 
10 
15 
3 
6 
10 
5 
4 
10 
20 
4 
10 
10 
5 
15 
5 
10 
6 
6 
1 
8 
6 
-3 
5 
-6 
-22 
10 
-21 
10 
6 
17 
