Properties of Pfaffians. 
207 
employed in reaching any case of the theorem, we now have two sets, the 
one arising from consideration of the rows and the other from the 
columns. For example, the determinant being 
1^,^36^465/61, or A say, 
we have for it the six expressions 
a^A^ + a^A^^ ... +aeA6, 
and the other six 
a,A, + b,B,+ ...+f,¥, 
a^A^ + b^B^+...+f^¥, 
and our procedure now is to subtract the sum of any number of the one 
set from the sum of any number of the other set. Thus, if in the first set 
the expressions summed be the 2nd and 3rd, and in the second set they 
be the 1st, 5th, 6th, the result of subtraction is 
e, 65 ee 
/x fs U 
6$A, A5 Ag) 
E, Ej Eg 
P. F; 
B, BJ. 
When the number of expressions taken from the one set together with 
the number taken from the other amount to 6, the arrays in the result 
besides being complementary are square : and when the number taken 
from both is 3, the coefficient of A is 0. 
4. The set of six expressions for ^ in § 2 is of course the analogue of 
the two sets of expressions for A in §3. The latter two sets, however, 
have associated with them two other well-known sets, namely, 
a,B, + a^B^ + . . . + agBg = 0\ 
a,G^ + a^G^ +...+^606 =0^ 
and 
a,A^ + b,B^ + ... + /,F3 = 0 
and when we seek for the analogues of these in the theory of Pfaffians, 
they are not so readily arrived at. It is interesting therefore to note that 
