180 
or sets of four vectors, accompanied each with another set of 
six; and the four or the six vectors in each set will have an ar- 
rangement among themselves, determined by the foregoing 
process; so that the 210 pyramidal and the 210 aconic func- 
tions have each a determined value, including a known posi- 
tive or negative sign or character. Each of the 210 products, 
thus obtained, is therefore itself also determinate, as being equal 
to some one positive or negative number, of which the sign as 
well as the absolute value can be definitely found, and may be 
considered as being known, before we introduce or employ any 
rule for combining or incorporating these various products 
among themselves, by any additions or subtractions. But if 
we now employ, for such incorporation, the rule that all those 
products which have been formed by any even number of bi- 
nary interchanges, from the product first assumed, which we 
may still suppose to be 
012345 . 6789, 
are to be algebraically added thereto; while, on the contrary, 
all which are formed from that original product by any odd 
number of binary interchanges are to be algebraically sub- 
tracted from it: we shall complete (as was before more briefly 
stated) the determination of that function of TEN vectors, 0 to 
9, which was lately called the apEUTERIC. 
Indeed, it may for a moment still appear that this function 
is in some degree indeterminate, because there may be many 
different ways of passing, by successive binary interchanges, 
from one given set of six, and a companion set of four vectors, 
to a second given set of six, with its own companion set of 
four. For example, we passed from the first to the tenth of 
the products already written, by a succession of nine binary in- 
terchanges, which may be indicated thus : 
56, 67, 78, 89, 45, 98, 87, 76, 57. 
But we might also have passed from the same first product, 
+ 012345 . 6789, 
