683 
1908-9.] Superadjugate and Skew Determinants. 
27. We have already used (§ 24) the fact that if A be an 'a-line skew 
determinant with univarial diagonal 
A - a n + a n ~ 2 .2 [«/5] 2 + a 71-4 . ^ [«/5yS] 2 + 
where a 8 is any pair of the lirst n integers, a(3yS any set of four, and 
so on. By adding to 2 the suffix —p to signify that a, (3, y, . . . are not to 
be taken from 1, 2, . . . , n, but from 1, 2, . . . , p — 1, p + 1, . . . , n, we can 
readily give expression to the like development of any primary coaxial 
minor of A : for example, 
An = an ~ l + a n ~ 3 .'^[afi] 2 + a n ~ 5 . y [a/jyg] 2 + . . . 
- -1 
With the same notation we have 
r i r i = ^ 2 + Zt la ] 2 ’ 
-i 
and therefore by multiplication 
— An.rp*! = a n -h a n ~' 2 j ^[ a 8Y + Z.D a 1 3 l 
a I -i -i J 
+ a n ~ 4 { Z l a Yy 8 Y + ZM^-Z^] 2 \ 
i -i -i -i ) 
+ 
Comparing this development with that of A, we see that on subtraction the 
coefficients of a n and a n ~‘ 2 would vanish, that the coefficient of a n_4 would be 
Z [ a /5yS] 2 + Z[ la P2W] 2 “ Z[« 
-i -i 
i.e. 2[ la ] 2 2t a ^] 2 _ Z[ la ^l 2 * 
-1 -1 
and that as a collective result we should have 
kj.r.-A = 
a I -i -i J 
1-1 -1 j 
+ (xxxiii.) 
28. The simplification, however, does not end here, there being connected 
with the coefficients on the right a rather remarkable theorem which 
enables us to substitute for each of them a sum of squares. 
To illustrate the nature of this, let us examine the coefficient of <x w-4 when 
n = 5. The first part of it is 
| [23] 2 + [24] 2 + [25] 2 + [34] 2 + [35] 2 + [45] 2 J | [12] 2 + [13] 2 + [14] 2 + [15] 2 | , 
