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 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 value (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 theirs* 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 



