( 695 ) 
NOTE ON THE SO-CALLED YAHLEN EELATIONS BETWEEN 
THE MINOES OF A MATEIX. 
By Sir Thomas Muir. 
(Eeceived November 3, 1915.) 
(1) Eelations between the elements of a rectangular array, though not 
looked upon as such, make their appearance very early in the history of 
determinants. In 1748, for example, Fontaine, using a peculiar notation 
for \ai h.,\, states in regard to the array 
«3 ^3 
\ h h h 
the identity 
|''l^2r;^3^4l — ki?>3|-|ao&^l + IC^Al^-.A^i = 0 . . . (a) 
In 1779 along with a series of related results Bezout gives in regard to 
the array 
^1 ^1 ^1 ^i/i 
a.2 h, c,^ d.2 e^fo 
H h e.J.^ 
the identity 
And, again, in 1809, Monge, in regard to the 3-by-5 array, 
d^ By 
ao c.) 
a,^ \ Co d^ 63 
gives five relations of the form 
1^162^31*1^1^2^31 ~ \oi-[b.2d.^\' a-^c.je.^\ + laj^./gj-la^c^c^gt =0 . • (7) 
each one of the ten minors of the array being involved three times. 
(2) Of course, these three identities were soon recognised as belonging 
to the same family, and were included in one general identity. To this 
result several writers contributed ; but the theorem is spoken of somewhat 
loosely as Sylvester's, and as giving an expression for the product of two 
w-line determinants in the form of an aggregate of like products. 
