184 
corners of a gauche hexagon aBCDEF, may now be concisely 
expressed by the symbol 
O.ABCDEF; 
or even simply by aBcpeEr, the reference to an origin being 
understood. To construct it, Sir W. Rowan Hamilton con- 
structs first the six vectors 
VAo, Vcaes veo a, V.a as Vigta’, Vit as 
and then the three other vectors #3, (’, 3’, which depend on 
these, in order to form thence that scalar S . 33'3", which, by what 
was stated near the commencement of the present Abstract, 
is the aconic function required. It will be seen that all the 
steps of the following construction of that function are in this 
way obvious consequences from the quaternion expression 
above given. The construction itself was communicated to a 
few scientific friends of his about the end of August and be- 
ginning of September, 1849, and has since been publicly stated 
at the Edinburgh Meeting of the British Association in 1850, 
although it has not hitherto been printed. 
Regarding the given and gauche hexagon, ABCDEF, as a 
sort of base of a hexahedral angle, of which the vertex is the 
assumed point o, Sir W. Rowan Hamilton represents the 
doubled areas of the six plane and triangular faces of this 
angle, namely, 
AOB, BOC, COD, DOE, EOF, FOA, 
by six right lines from the vertex, 
OL, OM, ON, OL, OM’, ON’, 
which are respectively xormals to the six faces, and are dis- 
tinguished from their own opposites by a simple and uniform 
rule of rotation: for example, the line ot contains as many 
linear units as the doubled area of the triangle aos (to the plane 
of which it is perpendicular) contains units of area; and the 
rotation round ot from oa to oB is right handed. The doubled 
areas of the three new triangles, 
