181 
by the two binary interchanges 47,56, to this other product 
and sign, - 
+ 012376 . 5489, 
where the sign + is prefixed, on account of there being now an 
even number (two) of such changes. On the other hand, the 
odd number (nine), of binary interchanges above described, 
had given the term 
— 012376 . 4589. 
But because, by the properties of the pyramidal function of 
four vectors above referred to, we have 
+ 5489 = — 4589, 
the two terms thus obtained differ only in appearance from 
each other. And similar reductions will in every other case 
hold good, in virtue of the properties of the pyramidal and 
-aconic functions, combined with a principle respecting trans- 
positions of symbols (which probably is well known) : namely, 
that if a set of n symbols (as here the ten figures from 0 to 9) 
be brought in any two different ways, by any two numbers / 
and m of binary interchanges, to any one other arrangement, 
the difference m —1 of these two numbers is even. 
The va.vuE (including sign) of the foregoing adeuteric func- 
tion, of any ten determined vectors, is therefore itself com- 
pletely determined, if we fix (as before) the arrangement of the 
ten vectors in the first of the 210 terms from which the others 
are to be derived: because the value of each separate term be- 
comes then fixed, although the forms of these various terms may 
undergo considerable variations, by interchanges conducted as 
above. If then we choose any two of the ten vectors, suppose 
those numbered 4 and 7, we may prepare the expression of the 
* adeuteric function as follows. We may first collect into one 
group the 70 terms in which these two vectors both enter 
into one common aconic function; and may call the sum of 
all these terms, Polynome I. We may next collect into a 
second group all those other terms, in number 28, for each of 
