1888 - 89 .] Dr T. Muir on the Theory of Determinants. 403 
(b - a)(c - a)(c - b) by PD (abc ) , 
(b - a)(c - a)(c - b)(d - a)(d - b)(d - c) by PD (abed ) , 
and .*. abc(b - a)(c - a) (c - b) by PD(0 abc). 
Lastly, he combines the two notations ; and any reader who 
remembers Cauchy’s mode of solving a set of simultaneous linear 
equations can with certainty predict the result of the union to be 
determinants. A new notation and a new name for the functions 
thus come into being together, the determinant of the system 
CL i CL 2 
being represented by 
£abcPD(abc) or £PD(0 abc) , (vii. 9) 
and being called a zeta-ic product of differences. (xv. 7) 
These special zeta-ic products being reached, the rest of the paper 
is taken up with an account of some of their properties, and the 
application of them to the discussion of simultaneous linear equations. 
Some of the matter may be passed over as being already familiar to 
us, although its earlier appearances were certainly made in a less 
picturesque dress. The first really fresh theorem concerns the zeta-ic 
multiplication of a determinant £PD (fdabc ... Q by those symmetric 
functions of a , b, c, . . . , Z, which we would denote by 
%a, 3 ah , 3 abc , .... 
but which Sylvester writes in the form 
S fabc . . . Z), S fabc . . . Z), S 3 (a&c . . . I ), .... 
In his own words it stands as follows (p. 39) 
“Let a, b, c, . . . I , denote any number of independent 
bases, say (n — 1); but let the argument of each base be periodic, 
and the number of terms in each period the same for every 
base, namely ( n ), so that 
