1899 - 1900 .] Dr Muir on the Theory of Shew Determinants. 191 
case, viz., where the elements are such that rs is always equal to 
- sr, there are only 1.3.5... (2 n - 1) different terms in 
X 
X, 
2n—l,2n ) 
that the aggregate of these is also got without repetition in a 
particular way already announced by Jacobi; and that it is this 
aliquot part of 2 ± A 12 A 34 . . . A 2w _i, n which constitutes ‘ la fonction 
de M. Jacobi.’ Jacobi’s theorem regarding the effect, on the func- 
tion, of interchanging two indices, is then restated; and a step 
further is taken in affirming that the function vanishes when two 
indices are equal. Finally, another law of formation — the recurring 
law — is given in the form 
[12 . . . 2 n\ = 12[345 . . . 2 n\ + 13[45 . . . 2t*;2] + 14[5 . . . 2tz,2,3] + . . 
which, of course, is in substance not different from Jacobi’s 
R = 
SR 
'da 
+ a , 
is 
0R 
2s da 0u 
The digression on ‘ les fonctions de M. Jacobi’ being exhausted, 
Cayley returns to skew symmetric determinants with the requisite 
It is instructive, in connection with the matter in hand, to note that this 
function is expressible in terms of four Pfaffians, viz., we have 
2 ± 12-34 = 2| 112 13 14 
- | 12 13 14 
1 23 24 
32 42 
34 
43 
+ | 21 31 41 
- | 21 31 41 
32 42 
23 24 
43 
34 
}» 
and thus see that, if the condition rs= -sr he introduced, the result is 
2 ±12*34 = 8* | 12 13 14 
rs= “ sr 23 24 
34 ; 
so that the Pfaffian on the right may be defined as the eighth part of a certain 
Cayleyan determinant ; or, in Cayley’s symbols, 
[1 2 3 4] = §2 ±12-34, 
where the 8 is the value of 2^ (1.2 .... \ri) when n— 4. 
Before leaving this it deserves to he noted that when Cayley came in 1889 
to re-edit his writings, he appended to this paper a note in which it is stated 
that part of his purpose was to show “that the definition of a determinant 
may he so extended as to include within it the Pfaffian ” (see Collected Math. 
Papers , i. p. 589). 
