432 PROCEEDINGS OF THE AMERICAN ACADEMY, 



show that the magnitudes are equal. The reversal of sign is to be 

 expected from the fact that the complement of a vector (whether 1- 

 or 2-) of class (7) is a (5)-vector (whether 2- or 1-), and vice versa. 



The use of the term complement in connection with scalars and 

 pseudo-scalars is sometimes convenient. Since, by the rule of inner 

 multiplication, we have ki24*ki24 = — 1, the complement of any 

 pseudo-scalar is a scalar of the same magnitude and of opposite sign. 

 We may define the complement of a scalar a as the product of the scalar 

 and the unit pseudo-scalar; thus a* = aki24. 



All the special rules for the inner products of unit vectors (and 

 pseudo-scalars) are comprised in the following general rule, which 

 will also be applied in space of four dimensions: If either of two unit 

 vectors has a subscript which the other lacks, the inner product is 

 zero; in all other cases the inner product may be found by so trans- 

 posing the subscripts that all the common subscripts occur in each 

 factor at the end, and in the same order, b}' then canceling the com- 

 mon subscripts, and by taking as the product the unit vector which 

 has the remaining subscripts (in the order in which they stand), pro- 

 vided that if the subscript 4 has been canceled, the sign is changed. ^^ 

 Thus 



ki24*k34 = 0, ki24*ki2 = k4i2'ki2 = kj, k]2*ki = — k2i'ki = — ko, 



ki24*k4 = k]2, ki34«kl4 = k3i4'kl4 = ks. 



30. Hitherto we have given little attention to the singular vectors 

 of our geometry, namely, the lines which are elements of a singular 

 cone and the planes which are tangent to a singular cone. We have 

 seen (§ 14) that the inner product of a singular 1-vector by itself is 

 zero, and have expressed that fact by stating that a singular line is 

 perpendicular to itself. Analytically expressed, the condition that 

 a 1-vector a shall be singular is that 



a-a = ar + tto- — 04- = 0. 



25 Instead of regarding the common subscripts as canceled, it is possible to 

 regard their corresponding unit 1-vectors as multiphed by inner multiplica- 

 tion, — - and in this case the change of sign takes care of itself. Thus 



Kpqr'Kqr = Kp {kg ' Kq) (Kr'KrJ- 



Indeed if a, b, C are mutually perpendicular 1-vectors, then all the rules given 

 above may be expressed in the equations 



(axb)'(axb) = (a«a) (b«b), (axbxc) • (axbxc) = (a«a) (b'b) (c«c), 



(axb)'b = a(b«b), (axbxc) -c = axb(C'C), 



(axbxc) • (bxc) = a(b'b)(c«c). 



