Pfa^ans connecied ivith the Difference-Product. '33 
The filial expansions are 
49(3^3 - 8coC, + 2OC1C5 - 120cq), 
13608(2c^ - 5C3C. + lOcoCfi - S^c^cj + 280cg). 
(7) At this stage we meet with a very curious relationship, the 
bracketed expressions just written being found to make their appearance 
in a totally different connection — namely, as playing a like part in the final 
•expansions of zero-axial skew permanents of the form 
In order to show this it is at least as easy to find the expansion of the 
more general permanent 
ay -\- + h.j 
a., a2 + h.y + 63 
a., 4- 61 a.^ + 60 a.^ + a.^ 
This we can express as the sum of 2" permanents with monomial 
•elements, the best mode of arranging them being in accordance with the 
number of columns of as and the number of columns of 6's which they 
•contain If the elements be all a's, the value of the permanent is 
n ! . a^a., ...an; 
if there be only one column of ?>'s the value is 
hr . (n — 1)1 ^a^ci.y...an-\, 
and the sum of all such is 
(n—l)\^h^.^a^a2.,.an.]; 
if there be only two columns of h's the value is 
hrbs . {n — 2) ! 2«i(^o...a;i-2, 
-and the sum of all such is 
2 ! {11 — 2)1 • 2«ia2---«"-2; 
-and so on, the last term, the {n -f 1)^^, being ^ 
n\ &i&2 --*^'" ' 
and in general the n^^^ term from the end being formable from the term 
;from the beginning by interchanging the as and' Vs. The expansion can 
therefore best be written in bifold form 
3 
