1904 - 5 .] 
Dr Muir on General Determinants , 
929 
Sylvester, J. J. (1851, March). 
[On the relation between the minor determinants of linearly 
equivalent quadratic functions. Philos. Magazine (4), i. 
pp. 295-305, 415 : Collected Math. Papers , i. pp. 241-250, 
251.] 
In order to formulate the relation referred to in the title, that 
is to say, the relation between any minor of the determinant (later, 
discriminant) of a quadratic form and the £ pari-ordinal ’ minors of 
the determinant of the new form obtained from the old by means 
of a linear substitution, Sylvester found it necessary to introduce 
“a most powerful, because natural, method of notation” for deter- 
minants. He says — 
“My method consists in expressing the same quantities 
biliterally as below : 
a 1 oq 
a 1 a 2 . . 
. agx n 
^2 a 2 * ' 
, . a 2 a n 
a n a- 1 
a n a 2 . . 
, . a n a. n 
where, of course, whenever desirable, instead of cq , a 2 , . . . , a n 
and cq, a 2 , . . . , a n , we may write simply a, b, . . . , l 
and a , j6, . . . , A respectively. Each quantity is now 
represented by two letters; the letters themselves, taken 
separately, being symbols neither of quantity nor of operation, 
but mere umbrse or ideal elements of quantitative symbols. 
We have now a means of representing the determinant above 
given in a compact form : for this purpose we need but 
to write one set of umbrse over the other as follows : 
fa 1 a 2 ... a n \ jf we now wish to obtain the algebraic 
\cq a 2 ... aj ° 
value of this determinant, it is only necessary to take 
a 1? a 2 , . . . , a n in all its 1*2 *3... w different positions, 
and we shall have 
j a \ a 2 ... a n i ^ 
\=2j ± { a i a ' 
\ cq a 2 ... a n ) 
0 ! X a 2 a 6>2 X 
X a n^ 6n } , 
in which expression _6 1 , 0 2 , . . . , 0 n represents some order 
PROC. ROY. SOC. EDIN. — YOL. XXV. 59 
