WILSON AND LEWIS. — RELATIVITY. 449 



axb =^ — bxa = {o\bi — oi/>i)ki.i + {ajji — a.ib>)ii2i + {a^bi — Uib^ijkst 

 + {(hbs — asb,)k-2:i + (03&1 — ai63)k3i + (01^2 — a.bi)kvu 



ax A = {aoAsi — asA-ii -\- OiA-nd^m + («3'Iii — (hAsi + UiAii) k3i4 



+ (ouhi — a^Au + a4^1i2)ki-j4 + (aKi-3 + a-iAv + f/a-li-j) k^a, 



axH = — Bxa = (oi^I^si + ai%n + a3^2li24 — a4^(i23)ki234, 



AxB = {AuB,3 + .-12453, + A^iBr, + Ar,Bn + ^31-B24 + .1,2^34) k,234. 



The outer product of two vectors the sum of whose (lin:iensions is 

 greater than four vanishes. The outer product of a vector by itself 

 vanishes except in the case of the biplanar or double 2-vector where 

 the product becomes 



AxA = 2(.-li4vl23 + ^24^31 + ^34^12) kl234. 



If the biplanar vector be written as A = B + C, where B and C are 

 two simple plane vectors, the product may be Avritten 



AxA = (B + C)x(B + C) = 2BxC. 



It thus appears that AxA is twice the four dimensional parallele- 

 piped constructed upon any pair of planes into which the double 

 vector may be resolved. The vanishing of the outer product, AxA 

 = 0, is the necessary and sufficient condition that A be uniplanar. 



The general rule for all cases of inner product has been stated (§ 29). 

 We may tabulate the following cases. 



a*b = ajji + 02'>2 + «3'->3 «4^4, 



a«A = (02^12 — 03.-I31 — a4-ii4) ki + (— 01.4,2 + a3.-l23 — 04.424) k2 



+ (ai.43i — a2.-l23 — 04.434) k3 + (— ai.4i4 — 0.0.424 — 03.434) k4, 



a-B = (0,3^(314 - «231l24) kl4 + (0,^)1,24 - 03?l,34) k24 



+ {a^lm - ai%u) k34 + (oi5r,23 - ai%2^i) k., 



+ (a25f)23 — 042^3,4) k3, + (a3'2tl23 — fl4'^(,24) k,2, 



A-B = - .414^14 — .124^^24 - .434^34 + .123^23 + .43,^3, + ^l,2i^.2, 



A-B = (- .42451,24 + .1345(3,4 + .l235(,23) k, + (.4,451,24 - .43451234 



+ .43,5(123) k2 + (- .4,45(3,4 + .(245(234+ .Il25(l23)k3 

 + (.4035(234 + .13,5(3,4 + .4,25(,24) k4, 



B06 = - 5(034^8234 - 5(3,4533,4 - 5(,2433,24 + 5(,23«,23. 



The geometrical interpretation of these inner products follows the 

 same lines as before. The inner product of a \ector into a vector 



