490 Proceedings of Royal Society of Edinburgh, [july 18, 
En indiquant done par S x des integrates qui supposent, dans 
chaque terme, les rnernes accens inferieurs aux lettres du 
meme alphabet, ces accens pouvant etre ou non les memes 
pour celles des alphabets differens, on pourra ecrire la 
precedente suite, en faisant usage de ce signe, ce qui donne 
S[S(y, OS(v,f)] = s ,[ZyvM-$zvZyQ- 
Cette nouvelle quantite est encore de la forme S(y', z"), en 
sorte qu’on peut dire que le produit de fonctions, telles que 
S{S(y, z') S(i>, 0} , 
sera lui-meme de la forme S (y, z").” 
This, if I understand it correctly, may he paraphrased and ex- 
panded as follows: — 
Take the product of two sums of s resultants, viz. 
toiVI + \v*W\ + feVI + •••• + 
x {KW + WV\ + h^l + •••• + Wl} 
or a \y}z?\ • 3 IV4 2 I, 
s=l s=l 
where, it will he observed, all the resultants in the first factor 
are obtained from the first resultant \yfzf\ by merely changing 
the lower indices into 2, 3, . . . , s in succession, and that the 
second factor is got from the first by writing v for y and £ for z. 
Then form all the like products whose first factors are 
I 2 / 1 VI, biVl> • • • ? \Vi n ~\ n \‘, 
these being along with \yfzf\ the \n{n- 1) resultants derivable 
from the two sets of n quantities 
Vi, Vi, Vi, • ■ • • , V* 
1 2 3 r; W' 
^1 > ^1 J # * 1 5 ^1 * 
The sum of these \n{n- 1) products may be represented, if we 
choose, by 
m<n 
Now if the multiplications be performed, there will be s 2 terms in 
each product, and the theorem we are concerned with has its origin 
in the fact that the sum of all 'the first terms of the products is 
