WILSON AND LEWIS. — RELATIVITY. 429 



axes will be denoted respectively by ki, kj, and k4. The outer products 

 kixko, ki^k4 k.vk4 will be denoted for brevity by kjo, k^, ko,. 



In terms of these arbitrarily chosen axes a 1 -vector may be repre- 

 sented as 



a = oiki + 02k2 + a^]lii. 



Similarly a 2-vector may be represented by the sum of its projections 

 on the coordinate planes as 



A = Aiikvi + .luku + -•l24k-j4. 



If we had chosen koi in place of kfi as one of our unit coordinate 2- 

 vectors, we should have written 



A = .l.-ikoi + .luku + A^^kn. 



Since A12 ki2 = A^x ls.21 and ki2 = — k^i, we have .^12= — ^21- 

 If we denote by ki24 the outer product kixk2^k4, then 



kl24 = ku2 = k4i2 = k42l = kijl = — kou, 



by the rules of outer products given above. In three-dimensional 

 space these products are unit pseudo-scalars. 



In terms of their components we may now expand the two types 

 of outer product which occur in three-dimensional space. In this 

 expansion we employ the distributive law and the law of association 

 for scalar factors. Then 



axb = (0162 — fl2&i) ki2 + (ajji — 0461) ku + (0264 — 0462) k24, 

 ax A = (ai^24 + 02^41 + a4.-lj2)ki24. 



At this point we may discuss the general characteristics of inner and 

 outer products of vectors of various geometric dimensionalities in an 

 n-dimensional space. In such a space we have vectors of 0, 1,2,..., 

 n-1, n-dimensions, designated as 0-vectors (or scalars), 1-vectors, 

 2-vectors, . . ., (n-l)-vectors, and 7i-vectors (or pseudo-scalars). The 

 outer product of a p- vector and a 5-- vector is a (j) + g) -vector; the 

 product vanishes if by translation the p- vector and 9- vector can be 

 made to lie in space of less than J) -\- q dimensions The inner product 

 of a ;j-vector and a g-vector, where j) ^ q, will always be defined as a 

 (p-9)-vector. Thus whereas the inner product of a 1-vector by a 

 l-\ector is a scalar, the inner product of a 1-vector and a 2-vector is 

 a 1 -vector. 



Both the inner and outer products will obey the distrii)utivc law, 

 and the associative law as far as regards multiplication by a scalar 



