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 note 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, to 

 9, which was lately called the adeuteric. 



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, 



