1917-18.] Roots of Determinantal Equations. 
57 
VIII. — A Determinantal Equation whose Roots are the Products 
of the Roots of Given Equations. By Professor William 
H. Metzler. 
(MS. received September 11, 1917. Read January 14, 1918.) 
Given two sets of linear homogeneous equations in n x and n 2 variables 
respectively 
Q=n\ 
(1) ']^ a fg x g = P • x f j (/ = 1 j • • •> n \) 
a= 1 
i=n 2 
(2) ^ liiVi • Vh ) (ll— 1, . . . , Wq). 
i = 1 
Let us form another set by multiplying every equation of (1) by every 
equation of (2). Thus 
or 
(A) 
where 
g=n 1 
%—n 2 
^ hiVi p • CT • Xj . , 
9=1 
i= 1 
'l = 71 o 
0=71! 
^PjPfhgi • w gi ~ ^ • w fh j 
g=i 
i — 1 
f~ 1, • • w x 
7^=1, ... , w 2 /’ 
PafiyS ^ay • ^a/3 ' *^a *27/35 and A p . 0". 
This set (A) of n = n x .n 2 equations is linear and homogeneous in the 
n = n 1 .n 2 variables w. It is of the same form as sets (1) and (2), and 
the determinant of it equated to zero is a determinantal equation in A 
that has for its roots the products of the roots of the determinantal 
equations in p and or, formed from equating to zero the determinants of 
\ 
(1) and (2) respectively, since A is the product of p and <7. 
If now we take another set of n 3 linear homogeneous equations 
3 =^. 5 
(3) (£=1» • • *5 n 3 ). 
5=1 
Let us jfrom (1), (2), and (3) form a new set by multiplying in all 
possible ways, taking one equation from each set. Thus 
g=m i=n 2 j=n 3 
cij-gXg . bhiVi . CfcjZ.j p . cr • t m Xf y frZf. , 
