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 otherwise 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 jirst 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 
