337 



The four systems differ onl\ by the different sigiiificatioii attaciied 

 to I and k. R'^ is the common vector-analysis, in which no difference 

 is made between polar quantities and axial ones and between vectors 

 and bivectors. R^ distinguishes between polar quantities and axial 

 ones. In Gibbs's form of this vector-analysis, owing to the groundlessly 

 introduced -f sign in 61.61= — 1, the formulae acquire apparently 

 irregular changes of -|- and — signs and the transvection-rule 

 becomes inetFectnal, so that the formulae stand side by side independ- 

 ent of one another and can be used only by means of a table. 

 When applied to units the rules for R's and R'^ are: 



ei X 62 = - 62 X 61 = 612 623 X 612 = 631 



ei 



ei=-l 



R"^ 61 . «23 = - I 



ei X ei2 = — 62 



61 1 = 1 •! = — 623 



R'. 



cycl. 1,2,3. 



cycl. 1, 2,3. 



(20) 



61 X 63 = — 60 X 61 : 



61 . 61 = — 1 



The rules (18) and (20) can be dualised according to all existing 

 dualities as given in the table. 



The free rules for Rl and Rl are: 



Transv. 

 numb: 



aXb = quantity of the second sub-degree 



1 a . b = scalar in k resp. 1 



aX(bXc) = aXbXc 



1 a . (b X c) = (a . b) c - (a . c) b 



a.(bXcXd)=^aXbXc.d = scalar in I resp. 1 



1 aX(bXcXd) = (a.b)(cXd) + (a.c)(dXb) + (a.d)(bXc) 

 1 a (b X c X d . e) = (a . b) (c X d X6) — (a.c) (bXdX6) + ... 



(a X b) X (c X d) = a X b X c . d 



1 (a X b) * (c X d) = (b . c) (a X d) — (b . d) (a X c) +. . . .') 



2 (a X b) . (c X d) = (b . c) (a . d) — (b . d) (a . c) 



1 (a X b) . (c X d X 6) = (b . c) (a X d X 6) + . . . 



2 (a X b) X (c X d X 6) = (b . c) (a . d) 6 + . . . 

 2 (a X b) (c X d X 6 . f) = (b . c) (a . d) (6 X f) + . . . 



2 (a X b X c) X (d X 6 X f) - (c . d) (b . e) (a X f) + . . . 



3 (a X b X c) . (d X 6 X f) - (c . d) (b . 6) (a . f) + . . . 



3 (a X b X c) (d X 6 X f . g) = (c . d ) (b . 6) (a . f) g + . . . 



4 (aXbXc.d)(eXfXg.h) = (d.6)(c.f)(b.g)(a.h) + ... 



(21) 



independent of the units used, viz. e^, e^, e,, 6< or 6^, e,, e,, e,. 

 ^) The index 2 under * is for simplicity omitted. 



