1917-18.] Roots of Determinantal Equations. 
59 
equation of (C) thus formed are not all distinct, and if we bring together all 
terms having the same variables there will remain as many distinct terms 
as there are terms in the expansion of the multinomial of m terms raised to 
There will also be the same number of 
| m f-k- 
-1 
1 
i 
-1 
distinct equations which we shall denote by (D), and for the eliminant in 
this case Muir (and later Pascal) has shown * that 
IV. 
R = A", 
where v = — 
m 
k j m + k - 1 
k 
m 
1 
m + k — 1 
m 
If, when the sets all become alike, we use different variables for each 
set, then it appears that from the way the equations (D) are formed 
the roots of the A-equation are the various terms, without the multi- 
nomial coefficients, of . . . -\-g m ) k , where cq, g 2 , ... g m are the 
roots of the A-equation of (A). For if we set the roots g 1: g 2 > • • • g m 
down k times forming k groups we see that the values of A are the 
products of the g s taken one and but one from each group.]- For example, 
if the two sets (1) and (2) become alike, then the determinantal equation 
of (C) has for its roots the squares of the g s and their products, two at 
a time, each repeated. The roots of the determinantal equation of (D) 
are the squares of the gs and their products, two at a time, without 
repetitions. 
Another simple proof that the roots are in this case the squares and 
unrepeated products of the g’s is furnished by starting with |C« — A | ,. 
where | C*i — A 1 = 0 represents the determinantal equation of (C), and by 
easy combinations of rows and columns break it up into the factors 
| Da— .A | and |A t j 2 — A, where A*/ is the element in the ith row and y th 
column of the second compound of the determinant A. It is known J that 
the roots of |A fi 2 — A | = 0 are the product of the c/’s, two at a time, and 
it therefore follows that the roots of ! D ti — A | = 0 are the squares and 
unrepeated products,, two at a time, of the c/’s. Thus, for the case m = 3 
and k — 2, we have for the determinantal equation of (C) 
* Muir, South African Assoc. Adv. Sci., vol. i (1903) ; Pascal, loc. cit., 1. 
Pascal in his Die Determinanten calls this the Scholtz-Hunyady determinant. It was 
considered by them in 1878-1880, but, as Muir points out in his paper, it was considered 
by Brill, Math. Annalen , Bd. 3, pp. 459-468, in 1871. 
f Metzler, Am. Jour. Math., vol. xvi, pp. 131-150 (1893); cf. Bados, Math. Annalen 
Bd. 48 (1897). 
J It may be observed that since R is the product of all the roots, we have another 
proof of IV. 
