58 
Proceedings of the Royal Society of Edinburgh. [Sess* 
or 
(B) 
where 
j=n, 
i—n<i 
U— n i 
2/ QfWij w gij = ^ • w fh!i J 
0 = 1 
i=l 
J = 1 
QafiySeY} ^aS • ^/3e • ^yr)j ^a/3y *®a • i//3 • ^y j cllld A p , CT, T. 
This set (B) of 71 = 71-^.712 >n z equations is linear and homogeneous in 
the same number of variables w. It is of the same form as (1), (2), and 
(3), and the determinant of the set equated to zero is a determinantal 
equation in A that has for its roots the products of the root of the deter- 
minantal equations of (1), (2), and (3), since A is the product of p, <j, r. 
(C) Continuing in this manner we may form a new set of equations 
from k linear homogeneous sets (1), (2), . . . (k) in n lt n 2 , . . . , n k variables 
respectively. The new set will contain n = n l .n 2 . . . n k equations in 
the same number of variables and the determinant of the set equated to 
zero is a determinantal equation in A whose roots are the products of the 
roots of the determinantal equations of the sets (1), (2), . . . k, since A is 
the product of p, cr, t . . . 
If we denote the determinants of the left-hand sides of (1), (2), . . . (k) 
by A, B, . . . K respectively, and that of the left-hand side of (C) by R, 
then R is the eliminant of the equations (C) when A is zero and it is 
known * that 
n n n 
I. R = . B^ 2 . . . K> . 
If n 1 — n 2 = n 3 = . . . = n k = m, then 
J J J> _ nJ c ~ 1 1 
If, further, all the sets become alike, except that the variables may or 
may not be made alike, then 
III. R = A*- m * -1 . 
(D) In this case when the variables are alike, the terms in any one 
* Metzler, Am. Math. Monthly , vol. vii, pp. 151-153 (1898-1900) ; cf. Moore, Annals 
Math., 2nd series, vol. i, pp. 177-188 (1900); Pascal, Rend. Girc. Mat. Palermo, t. 22, 
pp. 371-382 (1906). 
It may be observed here that, since B is the product of the roots of the equation in \ and 
is a product of powers of the roots of the equations in p, <r, r . . . , and therefore a product 
of powers of the determinants A, B, . . . K ; and since the determinants are in general 
irreducible and no one has prominence over another, we have another proof of I. 
The case of I for h — 2 is said to be due to Kronecker, and reference may be made to 
Jour. f. Math., Bd. 72, pp. 152-175 (1870). For other proofs, vide Hensel, Acta Math. y 
Bd. 14, pp. 317-319 (1891) ; Netto, Acta Math., Bd. 17, p. 200 (1893). 
