495 
1887 .] Dr T. Muir on the Theory of Determinants. 
are formed, the number in each new row being clearly 
i.e ., \n{n - 1) (n - 2) {n + 1) . 
The new quantities are, of course, not written by Binet in the form 
III!’ 
but the fact that they are resultants of the 2 nd order is carefully 
noted. Each of them is shown to be transformable, by a theorem 
which may be viewed as an extension of a result given by Lagrange, 
so as to have two of the elements resultants of the 3 rd order, and 
the others resultants of the 1 st order. This is done by taking, for 
example, the identities 
x fyM + y h \Wj\ + ^ = fjjjiZjl , 
XklVif + yjc\wA + ZkfiVA = Wyif > 
multiplying both sides of the first by x k) and both sides of the 
second by x h , subtracting, and writing the result in the form 
KVhW^A + \x k z h H^l = ocj c \x h y i z j \ - x h \3c k yiZj \ , 
x k 
f k yfzj\ 
WVi*A 
where of course it has to be noted that in many cases one of the 
resultants of the 3 rd order will vanish. The quantities, therefore, 
to be dealt with, are 
• • • , x A x hVi z A ~ x A x *Vi z A > • • • 
ViWj^A > • • ■ 
• • • , the I y h^i x j | - yfy^A > • • • 
Z 1 \ X l V2 Z A > • • ■ 
• > zA z h x iVA - z A z iMjA > • • • 
. . , % n\ X n-2y n-V^n\ • 
By raising each of the elements of the first row to the second power, 
taking the sum and simplifying, we could, we are told, show that 
the result would be 
3|a , 1 y 2 ^ 3 |‘ j . 
Very prudently, however, another process is chosen. It is recalled 
that the quantities in the third triad of rows are related to those in 
the second as those in the second are related to those in the first, 
and that consequently the required sum of squares of resultants is, 
by the multiplication-theorem itself, expressible as a resultant, viz., 
vol. xiv. 1/2/88 2 1 
