681 
1908-9.] Superadjugate and Skew Determinants. 
same time it has the advantage o£ bringing to light a series of Pfaffian 
identities. Thus, to take the case where n — 5 and p — 2, we have from (xxi.) 
A 2 i = a 2l .a ?> ~ vp\.a? + ( — a 34 f 5 — « 35 F 4 — a 45 1 3 ).a — F 2 F 4 , 
A 22 = cP 4- (a 43 2 + a 14 ’ + -}- « 34 " + ck 35 2 4- ^ 45 2 )a“ 4- F 2 2 , 
A 23 = ^03. a 3 — rqr^.a 2 4 - (< 3 q 4 F 5 + a i5 F 4 "F ^45-^1)*^ — F 2 F 3 » 
A 24 = a 2 r a 3 - r 2 r r a 2 + ( - a 13 F 5 + a 15 F 3 - a^F^.a - F 2 F 4 , 
A 25 = a 25‘ a ' i ” V 2 r 5' a2 "t ( ~ a i3 F 4 — ^14 F 3 + a 34 F l)-^ ~ F 2 F 5 
so that in 
A 2 ]Fi ~ A 22 F 2 + a 23 f 3 — A 24 F 4 4- A 25 F 5 
the cofactor of a 4 is — F 2 ; the cofactor of a 3 , as we see from § 22 (xxvii.), 
is 0 ; the cofactor of a 2 is 
rp 
■ F - 
{ a if "t ^14* 
+ * • • + V) F 2 - 
7 W ^ 3 
r 2 r 4-F 4 - 
r>r_p,.F r 
which, if increased by r 2 r 2 .F 2 — r 2 r 2 .F 2 , is seen from (xxviii.) to be 
= a 2 F 2 - r 2 r 2 . F 2 - (a 13 2 + a 14 2 4- . . . + a 45 2 )F 2 
= ~ ( r 2 r 2 ~ « 2 + a \f + a U + • • • + «45 2 ) F 2 
= - b 2 • 2 </>i 2 ; 
the cofactor of a 1 is 0 ; the term independent of a is -f 2 (F 1 2 +f 2 2 + . . . 
+ F 5 2 ) ; and the whole expression is 
- F,(<* 4 + a- . 20! 2 + 2<A., 2 ) , i.e. - F, . - . 
a 
It may be noted as a corollary that by putting p=l, 2, . . . , n in (xxx.) 
we obtain n equations from which F 4 , F 2 , . . . , F n may be eliminated, the 
result obtained corroborating (xxv.). 
25. The general property of Pfafhans which enables us to deal with the 
cofactors of the different powers of a in the foregoing method of proving 
theorem xxx. is that If F p F 2 , . . . , F n be the 'primary minors which 
pertain to the quasi- Pfaffian [123 ... n] when n is odd , and^ 
Mi i M 2 , Ms 5 • • • 
be used as abbreviations for the coefficients 
2ww. 2F^ a £l[ a £]> ••• 
of theorem (xxi.), then 
(P^-m j P^m J • • • j P'H'm $ 1 1 > ~ I 2 ’ F 3 J — 1 4 5 * • • ) “I” ( — 1 Y • PPm • F p 
^ 0 when m is odd. 
| \pa Y a 2 . . . a^^j] 2 when m is even. (xxxi.) 
In the left-hand member, it should be noted, there is no term contain- 
ing Fp, the means taken to indicate this being first to include a term 
