1907-8.] Dr Muir on the Theory of Skew Determinants. 309 
of two parts corresponding to A sr A and A rs A , the reason being the known 
existence * of a general theorem of determinants to the effect that if the 
product of | a n . . . a nn | and \b n . . . b nn | , obtained in row-by-row fashion, 
be | c u . . . | , then 
A rs • | 5^ .... b nn | ■ + ■ "1“ b ns G rn . 
This is seen to be immediately applicable on making | a u ... a nn | identical 
with A above and the b ’ s identical with the a’ s, and it, of course, implies 
that if the product obtained in column-by-column fashion be | c' u . . . c' nn | , 
then 
A sr ' | byi • . • • b nn | ^siC lr ■ d“ b sn G nr . 
Making the said necessary specialisations and noting that the two differently 
formed axisymmetric products are then identical (in other words, that 
C rs — C sr = C' rs = DgA, Torelli obtains 
i=n i=n i=n 
A„’A 
— a l s • 2 Alt + • ■ 
i= 1 
■ • + ^,2 A ri A si 4- • • 
i = 1 
i=n 
A ri A ni , 
i= 1 
i=n 
II 
cP 
2 
> 
+ 
• ■ + a ss ^ A r , : A si + • ■ 
i = 1 
d” Q'sn A r j A n i , 
i= 1 
whence by addition 
i—n 
(A rs + A sr ) A = 2a ss 2 A ri A *«- 
i= 1 
Cayley, A. (1865, Oct.). 
[A supplementary memoir on the theory of matrices. Philos. Trans. 
Roy. Soc. (London), clvi. pp. 25-35 : or Collected Math. Papers , v. 
pp. 438-448.] 
The expression of an even-ordered determinant, A 2m , as a Pfafhan being 
necessary for the second of the two investigations contained in his paper, 
Cayley effects the transformation (§§ 15-17) in substantially the same way 
as that devised by Brioschi ten years previously, the one point of difference 
being that the form of A 2m which is employed as a multiplier is got from 
A 2wi by reversing the order of the columns and then changing the signs of 
the elements in the last m columns. Thus, A 4 being 
abed 
e f g h 
i j h l 
m n o y , 
* Rubini’s Elementi d? Algebra (1866), p. 277, is referred to. 
