1899-1900.] Dr Muir On Jacobi's Expansion. 
139 
But to say that M, pq, rs, . . . , yz is a grouping of 1, 2, 3, , 
2 n into pairs, and that p is the number of inverted pairs in h k p q 
r s ... y z is exactly the same as to say that 
(~y.(a k -a h ) (a q - a p ) (a s -a r ) . . . (a z - a y ) 
is a term of the Pfaffian 
j a 2 -a 1 a 3 -a 1 .... a 2n —a 1 
« 3 — ' a 2 < V “«2 
O^n — a^ n -\ i 
where, he it observed, the suffixes of the two a’s in any element, 
are, when reversed, the place-numbers of the element. 
The theorem is thus fully established. 
(6) It is worth noting that in the general theorem used at the 
outset of the preceding demonstration the number of terms in the 
development is 
(mn ) ! 
m n . (n !) J 
and that this in the case where m = 2 becomes 
1.3.5. 7 ... . (M -1) . 2.4.6.,. 
2 n . 1.2.3.. 
or 
1 . 3 . 5 . 7 . . . . (2rc-l), 
which is, as was to be expected, the well-known expression for the 
number of terms in a Pfaffian of the n th order. 
Of course, in writing the various grouping of 1, 2, 3, . . . , 2 n 
into pairs it is desirable to write the members of each pair in ascend- 
ing order, and also to have all the first members of the pairs in 
ascending order. 
(7) On account of the co-existence of two rules for obtaining the 
same development, one of which is the rule for the expansion of a 
Pfaffian, it follows that the other rule, Jacobi’s, must contain 
within it the substance of the Pfaffian definition. 
