WILSON AND LEWIS. — RELATIVITY. 451 



The jjeneral relationships between profluets of veetors and their 

 complements have been developed in u previous section for a space 

 of any dimensions. It was there shown that (except 37) formulas 

 (34)-(39) for the expansion of all types of products involving 1-vectors 

 and 2-A'ectors would be true in higher dimensions, and this is true 

 even if the 2-vectors in\olved hai)i)en to l)e l)iplanar, l)eeause any such 

 vectors is the sum of two uniplanar vectors and the equations are 

 linear or bilinear in the vectors. Similar equations may, if occasion 

 requires, be developed for products involving 3-vectors. 



39. \Ve have not yet considered those vectors whose inner products 

 with themselves are zero. The case of the 1-vector, which is an ele- 

 ment of the hypercone, need not be treated again in detail. For 

 such a vector 



a-a = «r + ni' + as" — or = 0. 



A uniplanar 2-vector such that A*A = satisfies the conditions 



AxA = 2 {AuAiZ + ^124^^31 + ^■134^112) ki234 = 0, 



A'A = — ^14" — A-i^ — .134- + ^23" + Az-c + Ai'f = 0. 



Such a vector is obviously a plane tangent to the hypercone; for it 

 can be neither a (7)- nor a (5)-plane. The singular plane has the 

 same properties as in three dimensional space. The element of 

 tangency may be found as follows. If a is any vector, a* A is a line 

 in the plane A, and (a* A) 'A is a perpendicular line of the plane. But 

 the only line which is perpendicular to another line in this peculiar two 

 dimensional space is the singular line, that is, the element of tangency 

 with the hypercone. If Icj be taken as a, the element may be written 

 as 



(k4-AVA = kl (^34^131 — ^24^12) + k2 (^14^12 — -434^23) 



+ k3 (^24^423 - ^.4.431) + k4 {A,C- + A.:~ + ^-^34'), 



an equation which we shall find serviceable. 



The complement of a uniplanar singular 2-vector is itself such a 

 vector, and it may readily lie shown to pass through the same clement 

 of tangency. Indeed through every element of the hypercone is a 

 whole single infinity of tangent planes which are mutually comple- 

 mentary in pairs. 



If a 2-vector be biplanar, that is, if AxA is not zero, the condition 

 A'A = is satisfied when, if the vector be resolved into the two 

 complementary (7)- and (5)-vectors, these have the same magnitude. 

 For if 



A = ;/;M + //N, A«A = m- — n-. 



Such a vector is singular only in an analytical sense. 



