140 Proceedings of Royal Society of Edinburgh. [sess. 
This implied definition may be formulated as follows : — 
One term of the Pfaffian — 
| 12 13 14 .... 1, 2 n 
23 24 .... 2, 2 n 
34 .... 3, 2 n 
2ra-l, 2 n 
is 
12 . 34 . 56 (2n - 1, 2n) ; 
this is increased to three (1x3) by performing upon it the cyclical 
substitution 
/ 2n - 2, 2n — 1, 2n \ 
ll2n- l, 2n, 2n-2/; 
these three are increased to fifteen (1*3*5) by performing on each the 
cyclical substitution 
( 2n - 4, 2n - 3, 2n - 2, 2n - 1, 2n \ 
^s2n - 3, 2n - 2, 2n - 1, 2n, 2n - 4/ > 
and so on : all the terms being initially positive , but the sign of any 
one being changed as often as it is necessary to put the members of 
an inverted pair into their natural order. 
Thus the first term of the Pfaffian of the 3rd order 
13 
14 
15 
16 
23 
24 
25 
26 
34 
35 
36 
45 
46 
56 
is 12*34*56; and by the cyclical permutation of 456 we obtain 
two others 
+ 12*35*64 + 12*36*45 
or 
- 12*35*46 + 12*36*45; 
and lastly from these three by the cyclical permutation of 23456 
we obtain the remaining twelve terms. 
No other definition shows so clearly that the total number of 
terms in a Pfaffian of the n th order is 1 *3 *5*7 . . . (2w — l). 
