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phraseology, the scalar part of a quaternion would be the 

 arithmic part of a grammarithm ; and the vector part of a 

 quaternion would be the grammic part of a grammarithm. 

 In the form given above, of the general equation of cones of 

 the second degree, the six symbols, a, . . .a^, denote six edges 

 of a hexahedral angle inscribed in such a cone ; the six 

 binary products aa',...a^a, of those lines taken in their order, 

 are grammarithms, of which the symbols v. aa', &c., denote 

 the grammic parts, namely, certain lines perpendicular re- 

 spectively to the six plane faces of the angle; the three pro- 

 ducts 



v.aa'.v.a'"a'^ &c., 



of normals to opposite faces, are again grammarithms, of 

 which the grammic parts are the three lines j3, fd', j3", situ- 

 ated respectively in the intersections of the three pairs of 

 opposite faces of the angle inscribed in the cone ; and the 

 equation (1) of that cone, which expresses that the arithmic 

 part of the product of these three lines vanishes, shows also, 

 by the principles of this theory, that these lines themselves 

 are coplanar: which is a form of the theorem of Pascal. 



The rules of this calculus of grammarithms, or of qua- 

 ternions, give, generally, for the arithmic or scalar part of 

 the product of the vector parts of the three products of any 

 six lines or vectors aa', /3j3', yy', taken two by two, the fol- 

 lowing transformed expression : 



s (v. aa'.v. /3j3'.v. TtO = s . ayy'. s . a'iSjS' - s. a'yy'. s. ajSjS'; (3) 



and by applying this general transformation to the recent 

 results, we find easily, that the equation (1), under the con- 

 ditions (2), may be put under the form: 



S.aa'a" S.a"a"'a"' S.aa'a^ S . a^a"W 



(4) 



. aa a a . a a a s . aa a s . a a a 



which is another mode of expressing by quaternions the ge- 

 neral condition required, in order that six vectors a, . . . a^, 



