Mr. Herschel on circulating junctions , &c. 151 
by Mr. Babbage in the Philosophical Transactions foi> 1816) 
and let us consider the symmetrical function 
S 
4T— -M + I 
} 
Then, by the nature of the roots of unity as demonstrated in 
most treatises on algebra, we have 
= &c. and so on. 
If then x be changed to x-\-i, the above function will become 
/ {s x _ n+I ,$ s , S x ~ n + 2 } 
that is, by the nature of symmetrical functions, (the order in 
which the elements are operated on making no difference in 
their form ) 
/ 1 > S 4:-« + 2>^-«+x} 
the same as before. The function therefore remains unal- 
tered, while x changes to oc + i> and is therefore invariable. 
Hence we have in general 
/{ § *>Sr_„ =/{ °> °1 
This proposition may be extended to any number of func- 
tions of the form S^, Sj^, &c. and to functions symmetrical 
in different senses relative to each set of them : thus we have 
Si” 1 . s >>, ....S,w +I> sw ^ „ S f. 
&p) 
=/{ -0,1,0, &c.} 
the accents pointing out the different kinds of symmetry ob- 
served in the composition of the function relative to its several 
elements. 
