182 



which the two selected vectors both enter into the composition 

 of one common pyramidal function ; and may call the sum of 

 these 28 terms, Polynome II. And finally, we may arrange 

 (after certain permitted transpositions) the remaining 112 

 terms into 56 pairs, such as 



+ 012345 . 6789 - 012375 . 6489, 

 and 



- 012346 . 5789 + 012376 .5489, 



and may call the sum of these 56 pairs of terms, Polynome 

 III. ; the rule of pairing being here, that the two selected 

 vectors (in the present case 4 and 7) shall be interchanged in 

 passing from one term of the pair to the other, with a change 

 of sign as before. But when the expression of the adeuteric 

 has been thus prepared, it becomes clear that each of its three 

 partial polynomes is changed to its own negative, when the 

 two selected vectors are interchanged. In fact, each term of 

 the first polynome changes sign, by this interchange, in virtue 

 of the properties of the aconic function of six vectors. Again, 

 each term of the second polynome in like manner changes 

 sign, on account of the properties of the pyramidal function 

 of four vectors. And finally, each pair of terms in the third 

 polynome changes sign, from the manner in which that pair 

 is composed. On the whole then we must infer, that the sum 

 of these three polynomes, or the function above called the 

 adeuteric, Changes sign, without othemoise changing value, 

 when any two of the ten vectors on which it depends are 

 made to change places with each other : whence it is very 

 easy to infer, that this adeuteric function vanishes, when any 

 two of its ten vectors become equal. 



Now the aconic function is of the second degree, with 

 respect to each of the six vectors on which it depends ; while 

 the pyramidal function is easily shewn to be only of the first de- 

 gree, with respect to each of the four other vectors which enter 

 into its composition. Hence each of the 210 terms of the adeu- 

 teric rises no higher than the second degree ; and if we equate 



