Pfafjians co7mected tvith the Diference-Prochict. 
31 
Indeed a series of identities can be obtained in this way bv giving different 
positions to the units in the last rows of the multipliers ; and from this 
series can be deduced another, namely, in the case of the third order, 
(a - (a - yf a« 
7' 
has the equivalents 
3 I aO/yly2 
when s is 0, 1, 2, 3 respectively. 
(4) In the double-alternant theorem above used the index of the power 
to which each binomial is raised is 1 less than the number of variables in 
each of the two sets. When the index is the same as the number of 
variables, that is to say, when the determinant is 
I (a^ + \Y («o + l.y . . . (an ^ hnYi 
no useful outcome is obtained by 2^utting h,. = — ; for, when 7i is odd, the 
resulting zero-axial skew determinant being of odd degree vanishes, and 
when n is even the resulting determinant is not skew. 
When the index is 1 more than the number of variables, one case is 
unfruitful and the other not. For if n be odd the determinant 
I (ai + (ao + h^y' + ^ . . . (a, + + M 
is not skew, and if n be even it is l)oth skew and of even order. Calling the 
three alternants spoken of 
we may note the general conclusion that J)n;p for the suhstitution 
hr — — Or becomes the square of a Pfaffian when n is even and p odd. 
(5) The theorem that is available for our purpose in regard to J)n , n+\ is 
one that appeared in the Transac. B. Soe. S. Africa, iii (1912), p. 182. For 
71 = 4 it is 
1 
a^a^a.^a^ 
-10 
-10 
-5 
-5" 
— '^a^a^a.^ 
where ^J, ^1 stand for the difference-products of the a's and &'s respectively. 
When in this br is put = —a,, and the even-ordered rows have their signs 
changed, the right-hand member becomes, like T>4, ;5, zero-axial skew, and 
we deduce 
