178 
convenient to give the name of the Aconic (or heteroconic) 
function of those six vectors; because in the more general 
case, when they are not sides of any common cone of the 
second degree, this function no longer vanishes, but acquires 
some positive or negative value. 
One of the most important properties of this aconic func- 
tion is, that it changes its sign without otherwise changing its 
value, when any two of the siz vectors on which it depends 
change places among themselves. Admitting this property, 
which there are many ways of easily proving by the general 
rules of quaternions, and observing that the following func- 
tion of four vectors, a”, a", a", a™, namely 
S. (a rite a‘") fa" ~ a”™) Cite ss a) 
can be shewn to change sign in like manner, for any binary 
interchange among the vectors on which it depends, and to 
vanish when any two of them are equal; denoting also, for 
conciseness, the former function by 012345, the latter by 
6789, and their product by 
012345 . 6789; 
Sir W. Rowan Hamilton proceeds to form, by binary trans- 
positions of these figures, or of the vectors which they denote, 
from one factor of each product to the other, accompanied 
with a change of the algebraic sign prefixed to each such pro- 
duct as a term, for every such binary interchange, a system 
of 210 terms, namely, 
+ 012345 . 6789 — 012346 . 5789 
+ 012347 . 5689 — 012348 . 5679 
+ 012349 . 5678 — 012359 . 4678 
+ 012358 . 4679 — 012357 . 4689 
+ 012356 . 4789 - 012376 . 4589 
+ (a hundred other products) — (another hundred products) ; 
