1887.] Dr T. Muir on the Theory of Determinants. 
491 
expressible as a resultant by applying the multiplication-theorem, 
likewise the sum of all the second terms, and so on, the result 
being an aggregate of s 2 resultants. For if we fix upon a particular 
term of the first product, say the term 
\y h W\Wtj\ 
which arises from the multiplication of the h th term of the first 
factor by the 7d h term of the second factor, then take the corre- 
sponding term of the other products, and write down their sum 
\yfrh\-Wti?\ + \y h \ 
|V£* 3 | 
+ + | y,r% 
n — 1 
it is manifest that this sum is by the multiplication-theorem 
VhV 1 } + y h W + - • • • +Vh n v k n 
Vhtk + V 2 tk + • • • • + VhVh 
ZhW + ZhW + • • • .+z h n v t n 
Zh'U 1 + *k% 2 + • • • • + z h n L n 
Consequently since h may be any integer from 1 to s, and h like- 
wise any integer from 1 to s, the theorem arrived at is accurately 
expressed in modern notation as follows : — 
n = 2 
?n < n 
r s—s 
S| 
L s=i 
y, m z, n I • s’ | v,”L 
s—1 
1 
h=s VhW + VhW + • • • + ukv* + ZhW + • 
+ yft* 2 + • • • + y h n L n + • 
= 22 
7c=l A=1 
or 
y,} y k 2 . . 
k = 1 h=l 
Vh 
V V k 2 . 
& Ik 2 . 
Vk 
S' v 
4fc 
4- 7 n i> 7 
^ z h v k 
4-7 n r 7 
^T z h 4 k 
It is easily seen to be true of resultants of any order, as Binet 
himself points out. (xxx.) 
When s is put equal to 1, it degenerates into the multiplication- 
theorem. 
The theorem which follows upon this, but which is quite 
unconnected with it, may be at once stated in modern notation. 
It is — 
If ^|aqy 2 z 3 | denote the sum of the resultants obtainable from 
the three sets of n quantities 
/y» /y» ry* /y» 
cA /2 g • 9 • • 
Vi y» Vs • • • • y» 
h Z 2 Z 3 • • • • Z n, 
