1896 - 97 .] Dr Muir on Bordered Skew Determinant. 
353 
line of the Pfaffian of higher order is got by taking m and 
an even number of Id s. 
(4) that in the 2nd and 4th groups of Pfaffian products the 
Pfafhans are of like order, the second Pfaffian differing 
from the first merely in having k ’ s for Id s. 
(5) that in the 2nd and 4th groups of Pfaffian products the 
first line of the first Pfaffian is got by taking an odd 
number of Id s. 
Bearing these facts in mind, there will be little difficulty in writing 
out the development in any case. 
10. Knowing that the number of terms in the final expansion 
of a Pfaffian of the 1st, 2nd, 3rd, .... orders are respectively 
1, 3, 3-5, 3-5-7, .... we see that the number of terms in a bor- 
dered skew determinant, as given by the preceding development, is 
for the 3rd order, 
1+ 2-1 + 
1-3 
i.e . 
6, 
,, 4th ,, 
1+ 3-1 + 
3-3 +1-3 2 
i.e. 
22, 
„ 5th „ 
1+ 4-1 + 
6-3 + 4-3 2 + 1 ■ 3 2 *5 
i.e. 
104, 
, , 6 th , , 
1 +5-1 + 10-3 + 10-3 2 + 
5-3 2 -5 + l-3 2 -5 2 
i.e. 
576, 
7th „ 1 + 6-1 + 15-3 + 20-3 2 + 15-3 2 -5 + 6-3 2 -5 2 + l-3 2 -5 2 -7 2 i.e. 3832, 
so that the number of terms of a bordered skew determinant which 
cancel each other, and which we, by using the said development, 
are saved considering, are 
for the 3rd order 0 
2 
16 
144 
1208 
11. In connection with this, however, it has to be noted that 
we must not use the numbers 6, 22, ... . as being the actual 
numbers of unlike terms in the determinants in question, because, 
when we come to determinants of the 5th order, the terms of the 
expansion of (a 1 y 1 - a 2 /3 2 + a g/^) 2 make their appearance in the 
result, and these have to be counted as being 6 in number instead 
>> 
4th 
Ji 
5 th 
5) 
6 th 
J } 
5) 
7th 
>> 
. 
9 O 
. 
