673 
1908-9.] Superadjugate and Skew Determinants. 
sponding property of Cayley’s orthogonant. Following Torelli, we have 
from (xii.), on putting p — 1, 2, , n, 
a n(A-i 3 + A 2 i) + + A 22 ) + . 
• • “I - ®1 ni-h-nq 4" A q f) 
2 «A la ] 
%l(Ai<z + A f/ i) + a 2 2 (A 2g + A q2 ) + . 
• • 4“ 0>2 n(^-nq 4” A g ?i ) 
2aA 2 g 1 
a m( Ai q + A gl ) + a n2 ( A 2q + A q2 ) + . 
. . 4“ Ct/ nn ( A nq 4~ A gf 
'2aA ns , 
and solving we obtain, with the help of (xv.), 
-^pq 4~ A qp 2&(-A-ig , A 2 q , . . . , A n g $ A^ p > J ' * ' ) A np ) ~ A j 
which is essentially what was to be proved. 
13. If, instead of using (xii.) in the preceding paragraph, we were to use 
the altered form of it referred to in § 10, we should obtain for + 
another expression entitling us to assert that In the case of a shew 
determinant with a nnivarial diagonal the product of any two rows of the 
adjugate is the same as the product of the corresponding columns. (xvii.) 
In conjunction with this it is important to recall (viii.) ; also careful 
note should be taken of the complete diversity of the reasoning now 
employed to establish the same property. 
14. Taking the square of the adjugate of A, we have from (xvi.) 
(A"" 1 ) 2 = 
A 
11 
2(^12 + ^21) 
2(^12 + ^ 21 ) 
A 
22 
2 (Ain + A nl ) 
2 (A‘>n "4* A w2 ) 
2^A i n + A wl ) |(A 2m 4* A n2 ) 
whence, as Cremona noted, 
A \ tt 
a ) 
An + An 
A12 + A 21 .... 
A\ n 4” A nl 
-A- 12 4* A 2 i 
A-22 4“ A 22 .... 
A 2 n 4* A m2 
= (2a) n A n ~\ 
(xviii.) 
A in “1“ A nl 
A*2n "4" A n2 .... 
A . A 
- cx -nn _r r *-nn 
By dividing every row of the determinant on the left here by 2 a we 
obtain an expression for A” -2 , that is to say, for A, A 2 , A 3 , .... in the case 
of the 3rd, 4th, 5th, .... orders respectively ; and as these expressions for 
A, A 2 , A 3 , .... are not the ordinary expressions, there results by comparison 
a series of interesting identities. For example : 
3 
X 
a 
b 
— 
a 2 + x 2 
ab 
ac 
- a 
X 
c 
ab 
b 2 + x 2 
be 
-b 
- c 
X 
ac 
be 
e 2 + x 
X 
a 
b 
c 
2 
x 2 + d 2 + e 2 +f 2 
- bd - ce 
ad — cf 
ae 4 - bf 
- a 
X 
cl 
e 
- bd - ee 
x 2 4- b 2 4 - e 2 +f 2 
— ab - ef 
- ac 4- df 
-b 
-cl 
X 
f 
ad — cf 
— ab — pf 
v 2 4- a 2 + e 2 4- e 2 
-be - de 
- c 
— e 
-f 
X 
ae 4- bf 
- ae 4 - df 
- be - de x 2 
4- a 2 4 - b 2 4 - d 2 
vol. xxix. 43 
