185 



lol', mom', non', 



are next to be represented, on the same general plan, by three 

 new lines from the vertex, 



ol", om", on"; 



which three lines will thus be the intersections of the three pairs 

 of opposite faces of the hexahedral angle, and consequently 

 will, by Pascal's theorem, be situated in one common plane, if 

 the given hexagon abcdef can be inscribed in a cone of the 

 second degree, with the point o for its vertex. But in the more 

 general case, when the given hexagon cannot be so inscribed, 

 in any such cone with that assumed point for vertex, we can 

 construct a parallelepipedon with the three last lines, ol", om", 

 on", for three adjacent edges : and the volume of this solid is 

 the geometrical representation which SirW. Rowan Hamilton's 

 method assigns for what he calls (as above) the aconic func- 

 tion of the six given vectors, or of the six given points a, b, c, 

 d, e, f, in which those vectors terminate, or of the (generally 

 gauche) hexagon of which those points are corners. And with 

 respect to the sign of this function, it is to be regarded as 

 being positive or negative, according as the rotation round 

 on", from om" towards ol", is to the right hand or to the left. 

 Such then is the construction of the aconic function, 1 2345, 

 or abcdef ; and it is still more easy to construct the pyrami- 

 dal function 6789, which may also be denoted by the symbol 

 ghik; since the absolute value of this function is constructed 

 (as above remarked) by the sextupled volume of the pyramid, 

 which has the four points g, h, i, k for corners, or by the vo- 

 lume of the parallelepipedon which has gh, gi, gk for edges ; 

 while the quaternion expression assigned near the commence- 

 ment of this Abstract, admits of being thus written, 



S . (a 1 * - a VI ) (a vm - a") (a vn - a VI ), 

 and conducts to the regarding this volume, or the function 

 * 6789, or ghik, as being positive when the rotation round gh 

 from gi towards gk is right handed, but negative in the con- 



