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 six 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 VI , a TO , a™ 1 , a ra , namely 



S . (a* 1 - a vn ) (a vn - a™) (a™ - a 1 *), 



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) ; 



