Transactions of the Royal Society of South Africa. 
P(ag, a-, a^, a.., a^, a^, a^, a^, a.^, a_^, a^J 
a^ 
-«3 
«4 
-a. 
ao 
— a-j 
%~ 
a^- 
(Xo- 
— a-^ 
^3 
«4- 
a^ 
— % 
(Xo — 
<Xo — 
«3 
a^ 
a^ 
— 
— 
— ai 
wliich, on utilising the property regarding change of sign in the alternate 
diagonals of a pfaffian, takes the final form 
P (a(^„ a-, a^, a^, a-) 
= a.y 
-a^ a.^ — a,> a^—a.^. a- — a^ — a- 
a^ — (^\ ^'3~% ^-i — Of'3 — '^4 
a^—a^ (X;3 — a-o ^^4^ — (X.^ 
a.) — a^ 
a^ — a^ a.^ — a.2 a^ — ct.^ a^ — 
a^ — a^ a.^ — 
a2 — ft^ ^3 — 
a, 
«3 H 
^2 
a.y 
an 
(S) Equating the co-factor of (Tq on the one side of this to the co-factor on 
the other side we obtain 
i. e. P(«,-;, a^, ttg, 0.2, a.j, «), a^, a.^, a^) 
a^ a 3 a 2 
0.3 Wo «| <Xo 
C^o (X-j^ (X-j^ (X.T dfo 
(Xj ao (Xo o_j 
a^ ao a.^ a^^ a- 
— I (X'2-»i ^3~^3 <^'4~«'3 • j Ci2~^l ^3~^3 (Xj^-fto (^-f-Oij^ ^3 
ao-a-, a.T 
(Xj 
which is the analogous theorem in the case of an odd- ordered determinant. 
(4) In the second pfaffian which appears in both of the preceding results 
the last frame-line may be altered with impunity from 
f<^4 
^3 
ao 
. to - 
(X'O 
a^ 
a^i 
aj, 
for the reason that the fifth frame-line is the difference of the two. The 
latter form has the adv^antage of readily enabling us to equate the co-factors 
of a- in §2, and so arrive at the next lower case of the theorem, namely, 
P((X4, (X3, a.2, a^, tti, ao, a.^) 
a-i 
a^ 
a.y 
■a^ 
(Xo 
(X.J 
a-. 
a.y 
a.. 
a?, — (Xt —ac 
a,, 
■(X, 
— 
