450 PROCEEDINGS OF THE AMERICAN ACADEMY. 



of equal dimensions is a scalar, and is the product of either into the 

 projection of the other upon it. In the case where a biplanar 2-vector 

 is projected, or is projected upon, each simple plane has to be treated, 

 and the results compounded. That this may be done follows at 

 once from the distributive law. The product of two vectors of dif- 

 ferent dimensionality is a vector of which the dimension is the differ- 

 ence of the dimensions of the factors; this vector lies in the factor 

 of larger dimensions and is perpendicular to the factor of smaller 

 dimensions. However, the product a 'A, if A is biplanar, is com- 

 pounded of two 1 -vectors lying in the two component planes. 



The complement of a vector is again defined as its inner product 

 with the unit pseudo-scalar ki234. The complement of a 1 -vector is a 

 perpendicular 3-vector, and vice- versa; that of a simple 2-vector is 

 the completely perpendicular 2-vector. We may tabulate the results 

 for the unit vectors. 



-k4. 



With the aid of complements a unique resolution of a given 2-vector 

 into two completely perpendicular parts may be accomplished. Sup- 

 pose the resolution effected as 



A = mM + 7iN 



where M is a unit vector of class (7) and N one of class (5) so chosen 

 that MxN is a positive unit pseudo-scalar. Then 



A* = — 7iM + mN, 



mA — ?iA* „ nA + wiA* 

 and M = — —. — ^ » N = ., , ^ ' 



_-. - m-A — mnA* , n"A + nmA* 

 Hence A = — — 1 ^^ — -^ — • 



Let y = A'A = m" — n-, q = A'A* = — 2mn. 



The quantities m, n may then be expressed in terms of p, q, that is, 

 in terms of A* A, A* A*. The result is 



^^ i (Vp'^''+?+y)A + 9A* ^ , (Vp^+g^j-p)A-gA* 



