80 
Proceedings of Royal Society of Edinburgh. [sess. 
obtained by multiplying an element, f3' say, of the square array by 
the two elements ( rj , x) which lie in the same row and column 
with it but outside the array. The three determinants which are 
viewed as “ derivational functions ” of this function U are 
a fi .... 
a! (3' .... 
Ax + A'x + . . . B& + BV + 
R£ + S77 + . . . a (3 . . . . 
R £ + S >7 + . . . a (3 .... 
and 
R# + RV + . . . S# + S'#' + . . . 
+ Brj + . . . a f3 
+ B 'rj + . . . a /3' 
These are denoted by KU, FU, dU ; and the closing sentence of 
the introduction is, “ The symbols K, F, T possess properties 
which it is the object of this section to investigate.” 
KU, it will be observed, is what afterwards came to be called 
the discriminant of U; and FU, 1U are the results of making 
certain linear substitutions for the elements of the first row and of 
the first column of the determinant 
a y z 
( a p y 
rj a (3' y' ... . 
C « F y" — 
It is this determinant, therefore, which is under investigation and 
under comparison with U. That it is a bipartite function of 
x, y, z, . . . and £, y, £, . . . is manifest when we think of expanding 
it according to binary products of the elements of the first row and 
of the first column, the expression for it in the notation of 
