674 Proceedings - of the Royal Society of Edinburgh. [Sess. 
As in the latter example the non-diagonal elements on the right hand are 
- r x r 2 , 
- r x r 3 , - r x r 4 , 
“ r 2 ? 3 > ~ r 2 r 4 > 
(r q being used for the q th row), we should expect the diagonal elements to be 
~ r i r i » 
- V 2 r 2 » 
~ r 3 r 3’ 
- r 4 r 4 , 
i.e. - (x 2 + a 2 + b 2 + c 2 ) , - (x 2 + a 2 + d 2 + e 2 ) , . . . 
The explanation of the apparent anomaly is that the determinant which is 
squared on the left being equal to 
x 4 + x 2 (a 2 + b 2 + c 2 + d 2 + e 2 +f 2 ) + (af - be 4- cd) 2 
is not altered by the interchange 
fa b c\ 
-i / e d / > 
for if this change be made on the right we have only got further to alter 
the signs in the 1st and 3rd rows and then in the 2nd and 4th columns in 
order to evolve the familiar result. 
15. Denoting the p th row of A by r p and thej9 th row of the adjugate by R i; , 
we know (§ 11, xv.) that R^R^ is expressible as a determinant which is 
obtainable from A by deleting r q and inserting R^. As a result of multi- 
plying this determinant by A we obtain 
RpRq • A 
r l r l 
rp-2 
r 2 r i 
r 2 r 2 
7\r p .... ?\r n 
r 2 r p .... r 2 r n 
r q~ 1^2 
Wi 
t'q+d'o 
y y y y 
' q—V p • • • • ' q—v n 
A .... 
^ q+ dp • ■ • • ^ q-\-d n 
(-!)>'« A. (A (S A ( „), 
y y y y ^ p y y 
1 n' 1 7 iv 2 • • • • 7 n 1 p • • • • 7 n' n 
where A {p means the array got from A by deleting the p th row. But from 
(xvi.) we have 
R?;Rg 2^( Apg + Agp) } 
CL 
and thus by comparison there results 
A oAg = ( - ip +2 • KAps + Ki) - 1 
CL 
so that In the case of a skew determinant with univarial diagonal the 
product of any n — 1 rows by the same or any other n — 1 rows contains the 
determinant as a factor. (xix.) 
