| 
: 
; 
‘ 
179 
these remaining terms being easily formed in succession, ac- 
cording to the lately mentioned law. And to the algebraic 
sum of all these 210 terms, of which each separately is a po- 
sitive or negative number,—its positive or negative character 
depending of course not alone on the prefixed sign + or -, 
but also on the positive or negative characters of the factors of 
the product, which enters with that sign prefixed into the 
term,—Sir W. Rowan Hamilton proposes to give the name 
of the heterodeuteric, or (more shortly) the ADEUTERIC 
Function of the ten vectors a..a™, for a reason which will 
presently appear. 
To make the formation of this function of ten vectors more 
completely clear, it may be observed, that the function of four 
vectors, which has been above denoted by the symbol 6789, 
is easily found to represent the sextupled volume of the pyra- 
mid, whose corners are the terminations of the four vectors (all 
drawn from one common origin); this volume being regarded 
as positive or negative, according to the character (as right 
handed or left handed) of a certain rotation ; which character or 
direction is reversed when any two of the four vectors, and, 
therefore, also, their terminations, are made to change places 
with each other. On this account the lately mentioned func- 
tion of four vectors may be called their PyYRAMIDAL FUNCTION; 
and then the foregoing rule for the composition of the adeu- 
teric function may be expressed in words as follows :— Starting 
with any one set of four vectors, form their pyramidal func- 
tion, and multiply it by the aconic function of the remaining 
siz, out of the proposed ten vectors, arranging the vectors of 
each set in any one selected order. Choose any vector of the 
four, and any other of the six, and interchange these two vec- 
tors, without altering the arrangement of the rest, so as to 
form a new group of four vectors, and another new group of 
six; and multiply the pyramidal function of the former group 
by the aconic function of the latter. Proceeding thus, we 
can gradually and successively form all the 210 possible groups 
N 2 
