Pfaffians connected with the Difference-Product. 
35 
. . -36 -9ci 
. 84 . —9^2 Cg 
-9cq Cj 
10c«- 
Ot.-) — Cti 
tto — cti ao 
No direct mode of establishing them seems available, and the problem is 
not made easier by expressing them in terms of the differences of the a's. 
only : namely, for example. 
or, generally, 
ao — a^ 
— ttg ai — ag ttj — 
— 0-2 — ct^ 
— ag — 0-2 • ^3 ~ 
. (^2 - a2^)2'«+l 
= ( _ l)W»»-l).M.^i 
Cto — Cti 
— ^2 ^5^1 — ^3 
Cto — Clq 
+ 
0^2 — <^2w 
where the arithmetical multiplier M is 
(^2- (2^ + l)i (2^ + 1)2 
(2m + 1), 
(10) As the permanent on the right in the preceding is the same as that 
which occurs in the result obtained on making our usual specialisation in 
3)n ; «, we can eliminate it by division, and so obtain a relation between an 
axisymmetric determinant and a Pf affian ; for example, 
(a^-a^y . («^2-«3)*(%-a4)^ 
(ai-a^y(a2-a.^y . (a^-a^) 
(ai-a^y(ao-a^y(a~^-a^y 
= - (ai-a^y (ai--a,y (a,^a,y 
(ao — a^y (a^ — a^y 
(a^-a^y 
Similar procedure in connection with the theorems of §§1, 9 gives the 
quotient of two Pfaffians in terms of a zero-axial skew permanent. 
Note should also be taken of the theorem derived after the manner of 
§2, the lines of units now extending, however, to the permanent on the 
right ; for example 
