670 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the two expansions thus reached differing only in the signs of the terms 
which involve zero-suffixed determinants of odd order. The consequence 
is that for equality of A and A as a condition we may substitute the vanish- 
ing of the sum of the said terms. Hence The super adjugate of A will be A n 
when and only when 
2/ I ffii a 22 a 33 lo ‘ ft 44 a 55 a 66 • • •) + I ^11^22^33^44^55 lo * a 66 • • • ) + • * * = 0 . (iv.) 
Thus, the superadjugate of | a 1 b 2 c 3 1 is | a 1 b 2 c 3 1 3 if 
l a A c sl 0 = 0. i.e. if a 2 b 3 c 1 + ci z b Y c 2 = 0; 
and the superadjugate of | af} 2 c 3 d± | is | a-fb 2 c 3 d± | 4 if 
I C 1 ^ 2 C 3 lo ^4 + I lo C 3 + I | 0 ^2 + I ^2 C 3^4 loffi = 6 > 
i.e. if 
(a 2 ^ 3 c 1 + af> Y cf)d^ + (ajb^d x + a A b l d 2 )c 3 + (a 3 c 4 d 1 + a A c 1 d 3 )b 2 + (b 3 c A d 2 + b 4 c 2 d^)a l = 0 . 
5. Since the p th row of | a L1 a 22 . . . a nn | is 
O' pi > C^‘2 j . . . , eipp , • • • , &pn > 
and the p th column is 
®\p 5 ^2 p j • • ■ > ^pp j • • • j ® np ) 
it follows that In any shew determinant the sum of any row and the cor- 
responding column is twice their common element , (v.) 
or, say, 
row p + col i( = (0,0,..., 2 a pp , 0 , . . . ) . 
From this we can prove that If A be skeiv, any element s pq of the super- 
adjugate of A is 
(rowy of adjug. of A $ col g of A). (vi.) 
For from (v.) we have 
(row 2 of A + col 3 of A$row^ of adjug. of A) 
(0,0,..., , 0 , . . . $ A^ , Api , . . . , A pn ) 
^dqqApq , 
and from the fundamental property of the adjugate 
(row g of A $ roWp of adjug. of A) = 
therefore by subtraction 
(eol, of A § row,, of adjug. of A) = { f f P * _ 
0 if P=¥q, 
A if p = q ; 
as was to be proved. 
6. From (vi.) it follows immediately that the superadjugate of A, when 
A is skew, is the product of A by its adjugate, and therefore is A n as we 
have already seen in (iii.). Further, we have only to think of the case of 
