ROTATIONS I\ HYPERSPACE. 653 



protluct of more than two 1 -vectors. The dot product of two 1- 

 vectors is defined as the projection of one on the other multipHed by 

 the length of the one on which the projection is made. This agrees 

 with the Gibbs definition of the dot product of two 1 -vectors. Thi§ 

 product is a scalar. The dot product of two vectors of higher dimen- 

 sion is defined as the vector in the larger space perpendicular to the 

 smaller. The magnitude is equal to the magnitude of the projection 

 of the smaller into the magnitude of the larger. If the two vectors are 

 of equal magnitude this product is again a scalar otherwise it is a vector. 

 This definition differs widely from the usual definition of inner product. 

 This product is commutative while the ordinary inner product is not. 

 We shall choose unit vectors along mutually perpendicular axes 

 for our reference system. Let these reference vectors be ki, k2, . ■ ■ ■ kn. 

 Then the coordinate planes, 3-spaces, 4-spaces, etc. are 



kl2 = ki>^k-2, A'i3 = A'iXA"3, .... krs = kr^ks 

 k]23=^ kiXk^^ki, . . . .kpqr = kpXA^gXA'rj 



kpq . . . . r — k p^kgX .... x^v. 

 The dot product of these unit vectors are as follows: 



ki-ki = 1, ki-kj = 0, (i ^ j), 



kij-kij=l, kij-kii = 0, (j ^ I), kij-krrtn = {i,j 9^ m, n) 

 1-..1-.. — — h. I'-.l'- — I-- 



ki' k iji=^ kji, kif'kiji ^ ki etc. 



The dot product of two of these unit vectors vanishes if the smaller 

 is not entirely contained in the larger, that is if a subscript appears 

 in the smaller which does not also appear in the larger. If the dot 

 product vanishes the two vectors are perpendicular. This however 

 does not require complete perpendicularity, that is it requires that 

 one vector in one is perpendicular to the other while complete per- 

 pendicularity requires that every vector in one is perpendicular to 

 every vector in the other. To obtain a vector completely perpendicu- 

 lar to a given vector we must resort to the complement, which for 

 unit vectors is defined as the vector obtained by taking the dot product 

 of the given vector with the pseudo scalar.^ Throughout this paper 



5 The pseudo scalar is defined as the cross product of all the 1-vectors 

 arranged so that the value is unity. Thus in 4-space the pseudo scalar is 

 kiXkiXkiXki = kmi- 



