(29) 
NOTE ON PFAFFIANS CONNECTED WITH THE DIFFERENCE- 
PEODUCT. 
By Sir Thomas Muir, LL.D. 
(Eead April 19, 1916.) 
(1) In the year 1886 Torelli announced* without proof the identity 
(a., - a;3)2'»-i . . . (ao - a2„0^"'~^ 
(a2m-\ - «2«)^"'" 
= (_ 1)^»K.. + ]). (2m-l)i (2m-l)o . . . {2m- l),„-i.\al a] • • • a^^"~'\ ' 
where (2m— 1)^, (2m— 2)0, . . . are the coefficients of the expanded 
binomials. Probably the easiest way to establish it is to make it dependent 
on Zehfuss' double-alternant theorem 
1(0^1 + &i)«-i (a. + hoy-^ . . . (an + Ky-^\ 
Putting in this n = 2m, br — — Ur the determinant on the left becomes 
zero-axial skew, the binomial-coefficients become even in number with the 
first half the same as the second, the two difference-products become 
identical save as to sign, and the additional sign-factor thus introduced is 
the same as that already there. The extraction of the square root is thus a 
simple matter. 
(2) As a first step forward from this let us examine the consequence of 
putting in it a2m = 0. On the right-hand side the determinant, whether we 
view it directly as such or indirectly as the difference-product, is seen to be 
naturally separable into two parts, namely, 
(— iyyn-\a^ ao...a2m-i and | «i «'2---«2l-i | ' 
but on the left-hand side there is no corresponding appearance of resolva- 
* Giornale di Mat., xxiv, p. 377. See aJso Muir's textbook (1882), pp. 206, 240, 
ex 15. 
