788 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Multiplying by B 



(X + 1) AB = XB. 



Multiplying by CD 



(X + 1) AB • CD = XB CD 



= XB (CX + XD) 



= (XBD - XBC) X 



= (BCD) X. 



Again multiplying (40) by CD 



(X + 1) ACD = X (BCD). 



Using the last two relations with (40) 



(X + 1) A = ^^^^- B + (X + l)-^cBDy 



From which we get 



AB CD = (CDA) B - (CDB) A. (41) 



Starting with the relation 



(X 4-1) C = X + XD 



the following relation is obtained : 



AB CD = (ABD) C - (ABC) D. (42) 



These last two relations are the general formulae for Grassmann's 

 regressive product. Similar formulae can be obtained for the regres- 

 sive product 



ab • cd. 



Subtracting (41) from (42) we get 



(ABC) D - (ABD) C + (ACD) B - (BCD) A = (43) 



for the identical relation connecting four points of the plane. 



Scalar Products. 



47. The point F was exceptional only in the addition of lines. It is 

 lines through this point that are exceptional and not the point itself. 

 The preceding product theory then holds when F is one of the quanti- 

 ties involved. If A, B and F are unit points not on f 



(ABF)=AB + BF-fFA. 



