672 
Proceedings of the Royal Society of Edinburgh. [Sess. 
This is supplementary to (i.) and can be proved in the same way. From 
the two theorems it follows that, when A is skew and has a univarial 
diagonal, the product of the p th row of S by the q th row of A differs only in 
sign from the product of the corresponding columns if p and q be different, 
and does not differ at all if p and q be the same. 
10. If | a n a 22 . . . a nn | , or A say, be a skew determinant with univarial 
diagonal, the product of the p f/ ' row of A by the sum of the q f/l row , and 
cf column of the adjugate of A is 2aA pq . (xii.) 
This theorem is due to Torelli (see Giornale di Mat., iii., 1864, pp. 7-10). 
It may be written : 
a. 
'p\ ? 5 
Q’pn $ 
Ag2 "t" A 2q J 
A A \ _ 
■‘^■qn ~ nq ) 
2aA 
■pq 5 
and as, because of the skewness of A, 
(c#i Ci p , cl^p , ... , cip n "1 ct np $ A q i + A , A q2 1 - A 2q , ... , A qn + A nq ) 
— 2dpp(A.qp + hpq) 
it follows bv subtraction that 
(cj p j ct^p , , &np $ A ql + A q i , A q2 “I - A 2q , ... , A qn + A„ q ) 2cA q ^ , 
in other words, if for “ p th row ” in (xii.) we substitute “ p th column,” we must 
at the same time alter 2aA pq into 2aA qp . 
Further, since from (v.) we have 
(Ap! j Apq , • • • A p n $ CLqi *1" U,\q , Clq.y A C 2q , . . . , Ct qn "1“ d'nq) -‘^ , qqA pq , 
there results by comparison with (xii.) the curious fact that 
(ctpi , Ct p 2 ) . • • 5 $ A q i + A i q , A q2 + A 2 q j • . . , A qn + A n f 
is not altered by interchanging the as and As. (xiii.) 
11. It will be convenient to insert here two properties of general 
determinants, for which no proof is necessary. The product of the p 77i row 
of the acljugate of an n -line determinant A by any n elements whatever is 
the determinant obtained from A by substituting for its p 77 * row the said 
new elements. (xiv.) 
The product of the \) th and c( h rows of the adjugate of a determinant A 
is obtained from A by substituting for its p 77t row the q 77t row of the adju- 
gate, or for its q 77i row the p 77i row of the adjugate. (xv.) 
12. If | a u a 22 . . . a nn | , or A say, be a skew determinant with univarial 
diagonal, then 
(Api , A p , 
J Ap n $ A q i 5 A q2 , ... , A^ ?i ) 2 ( Apq I* Agp) 
a 
'whether p and q be different or the same. (xvi.) 
This was originally announced for the case where a = 1 by Spottiswoode 
in 1853 (see Crelle’s Journ., li. p. 261) as a deduction from the corre- 
