448 PROCEEDINGS OF THE AMERICAN ACADEMY. 



are tangent to the hj'percone, there are two classes of planoids, namely, 

 (5)-planoids which contain a (6)-line, and (7)-planoids which contain 

 no (5)-line. In the (7)-planoids the geometry is the ordinary three 

 dimensional Euclidean geometry; in the (5)-planoids the geometry 

 is that three dimensional non-Euclidean geometry which we have 

 discussed at length. 



Every (5)-line determines a perpendicular planoid of class (7), and 

 every (7)-line determines a perpendicular planoid of class (5). Thus 

 if we construct four mutually perpendicular lines, one will be a (5) -line, 

 and three will be (7)-lines. A plane determined by one pair of these 

 four mutually perpendicular lines is completely perpendicular to the 

 plane determined by the other pair, in the sense that every line of 

 one plane is perpendicular to every line of the other, and the planes 

 therefore have no line in common. In general every plane determines 

 uniquely a completely perpendicular plane. One of the planes is a 

 (7)-p]ane and the other is a (5)-plane. 



As in our previous geometries, perpendiculars remain perpendicular 

 during rotation. If then in a rotation any plane remains fixed, its 

 completely perpendicular plane will also remain fixed; and a general 

 rotation may be regarded as the combination of a certain ordinary 

 Euclidean rotation in a certain (7)-plane, combined with a certain 

 non-Euclidean rotation in the completely perpendicular (6)-plane. 



38. Let ki, k2, ks, k4 be four mutually perpendicular unit vectors 

 of which the last is a (5)-vector. The six coordinate 2-vectors may 

 then be designated ^^ as ki4, k24, k34, kos, ksi, kjo. There are furthermore 

 four coordinate unit 3-vectors k-234, ksu, ki24, kios; and a unit pseudo- 

 scalar ki234- We may represent 1-vectors, 2-vectors and 3-^'ectors, 

 as the sum of their projections on the coordinate axes, coordinate 

 planes, and coordinate planoids. Thus 



a = 0]ki + a2k2 + a^ks + 04k4, 



A = Jl4k|4 + ^l24k24 + .•l34k34 + .•l23k23 + A^ikn + Ar^^,' 

 B = ?l234k234 + ?(314k314 + 5(i24kio4 + ?ti23ki23. 



The outer product of any two vectors is defined geometrically and 

 expressed analytically in a manner entirely analogous to that of the 

 simpler cases already discussed. We thus obtain the following equa- 

 tions for the different types of products. 



32 The particular order of subscripts is chosen for convenience only. 



