336 



(first) vectorial product x 



second ,, ,, 2 



P 

 (only for p even) «-tli middle product, i( = -f 1. 



second scalar product 2' 

 first scalar product 

 With this notation, which is in agreement with the existing dualities, 

 products that are identical inith the rotational group obtain the same 

 name and the same si/mbol. Owing to the identification of I and k 

 with common numbers the first middle-product is identical with the 

 product of ordiruxry numbers mutually and with othei- {|Mautities, 

 hence its symbol may be omitted as l)eing customary. 



The rule of transvection. 

 If each factor is an alternating product of fundamental elements: 



;/a = ai .... dip' 



,/'b = bi . . . . b,^' 

 we can form the combination : 



(a,/ . bi) (ay/ 1 . b-.') • . . . (a// /-j-i . b, ) ai . . . . a/-) b,-f 1 .... b-,/, 

 repeat the- same for all p ! resp. (f ! modes of notation of ^/a and /,'b 

 and add the results. 



The sum then consists of p' ! q' ! terms, equivalent to each 

 other in groups of (/>' — /).' {q — i)! H. This sum divided l)y (// — i)! 

 {q — i)! a, or, stated more briefly, the sum of (///) (</'/) i.' ai'l)itrary 

 di^event terms, is called the i- f old-combination of y, a and ^'b. The 

 f-fold combination is now equal to the product with the transvection- 

 nnuiber i. The transvection-number of a product being known, we 

 can hence write it down from memory by this rule. 



The free rules for II ^ and R^. 



Hence the free rules for R^, R'l, Rl, Rl and R'^ are: 



Transv. 

 numb.: 



aXb= quantity of the second sub degree. 



1 a.b= scalar in k resp. 1. 



a.(bXc) = aXb.c= scalar in I resp. 1 . ^) 



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

 1 a(bXc.d) = (a.b)(cXd) + (a.c)(dXb) + (a.d)(bXc) ' (18) 



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



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



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



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



') In alternating products the brackets have been omitted for the association 

 (..)., so that we write the alternating product of aj, . . . ., a^y : 



ai X a2 X .... X an X an'-\-i . a,/-|-2 a ;,. 



