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corners of a gauche hexagon abcdef, may now be concisely- 

 expressed by the symbol 



o. abcdef; 



or even simply by abcdef, the reference to an origin being 

 understood. To construct it, Sir W. Rowan Hamilton con- 

 structs first the six vectors 



V.aa 1 , V.cfa 11 , V.a n a m , V.a ra a IV , V.a w a v , V.a v a, 



and then the three other vectors /3, /3', 0", which depend on 

 these, in order to form thence that scalar S . /3j3'j3", 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 normals to the six faces, and are dis- 

 tinguished from their own opposites by a simple and uniform 

 rule of rotation : for example, the line ol contains as many 

 linear units as the doubled area of the triangle aob (to the plane 

 of which it is perpendicular) contains units of area; and the 

 rotation round ol from oa to ob is right handed. The doubled 

 areas of the three new triangles, 



