685 
1908-9.] Superadjugate and Skew Determinants. 
Similarly we have 
Zt la2 ]"Z[ a ^ e fl 2 - Z^/W 7 ?] 2 
-i -i , -i 
= Z | [la][arstuv] + \l/3][/3rstuv]+ ... j> (xxxvi.) 
where on the right a, f3, y, ... is a set of n — 6 integers forming with 
r, s, t , u, v the full set 2, 3, ... , n ; and so on. 
The general proposition thus derivable regarding quadrate numbers is 
that The product of the sum of n — 1 squares by the sum of C n _ 1> 2m squares 
is expressible as the sum of C n _ li2m+ i + C n _ li2m _ 1 squares. (xxxvii.) 
30. Returning to the result of § 27 and making the substitutions now 
possible, we have the important theorem that If | a n a 22 . . . a nn |, or A n 
say, be a shew determinant with a univarial diagonal, then 
” A ir r i r i “ • Z { [ la l ar ] + \. 1 PlP r ]+ ••• } 
+ a n ~ & . 2 | [la][ars£] + [\/3][f3rst] + . . . | 
4 a n_8 . ^ | [la][a rstuv] 4 [1 [3][/3rstuv] 4 . . . j 
+ 
where, under the first 2, r is any one integer taken from 2, 3, ... , n, and, 
a, /3, ... . are those remaining ; under the second 2, rst is any set 
of three integers taken from 2, 3, . . . n, and a, /3, .. . are those remaining, 
and so on. (xxxviii.) 
Since the number of integers a, /3, .. . under the first 2 is n — 2, under 
the second 2 n — 4, and so on, the number of terms in the expressions to be 
squared in any case is 2 more than the index of the attached power of a. 
When n is even the last term is independent of a, and is a sum of squares 
of binomials for which there is an alternative mode of expression. This is 
due to the fact that the cofactors of [la], [1/3], . . . are then primary minors 
of [123 . . . ri\ and can be denoted by A 12 , — A 13 , A 14 , — A 15 , . . . , the term 
thus being 
( a i2 A I3 ~ **13^-12)" 4- (^12 A 14 — tt 14 A 12k + ■ • • 
For example, putting n — Q we have 
— An.nr, - A, = x 
.2 f (a. 
>9 ) 
a ui-'l'l ‘-*6 — ^ 1 ( r i r 2)“ "t • • • + dvtk [ + 
a i2 a l3 • • 
• a \6 
Ai 2 A ]3 . 
■ A 16 ! 
When n is odd, the last term is 
{ [!«][««* • • • J 
