JoLY — On the Place of the Ausdchnunsleghre. 15 



groups in the sub-products -Wi-stz . . . -sr,„ and -sr„j+i . . . ar„. In other 

 words, in the above expression F^ and Vn-m are superjiuous symbols. 



(5). We may replace the expressions by 



Pihh • • • L 

 and r^^iXoXs . . . X„ 



where l2= p^ - Pi, &(i., and A2 = -ro-o - -zti. 



(6). We can, by assuming the origin to be arbitivary, suppose the 

 vector -sTi to be the symbol of a point. ^ We have Ao = -2^3 - -ztj a 

 vector equal to that between the points, and Vz-sxiX^ we may interpret 

 as a point vector ; also the interpretations of V^Tr-iX^^i, &c. are obvious. 



We may, in forming the product, replace w^ by t-wi + rjz where 772 is 

 at right angles to -zzri, we may put t-^ = t'-ari + frj^ + rj^, where 773 is at 

 right angles to -ieti and -zzto or to -wi and 772, and we have iinally 



F^,-5ri-sr27B-3 , . . -nr^ = ^iTjorji - ■ . rjn, 



the symbol V„ being unnecessary on the right-hand side, as the product 

 is irreducible. 



Prom this follow all Grassmann's conceptions of the continuous 

 products of points. The product of two is the point-liue joining 

 them ; the product of three the point-plane determined by their 

 completed parallelogram ; the product of four the point- volume of the 

 completed parallelepiped determined by the four points. 



An apparent break in the parallelism occurs when the order of the 

 product equals the number of the units. Grassmann defines the 

 products of this order to be scalars. In our case Vjy'do-iizr.i . . . vTjy ^^ 

 equal to a scalar multiplied by O (= i^i^ . . . ij^) the product of all the 

 units. 2 Por present purposes we need not at all inquire into the 

 nature of fi. We are entitled to identify where n is less than iV, a 

 Grassmann " product " with Vn-ur-^vT^ . . . -zzr„ ; and where « equals N 

 we may identify a Grassmann product with the scalar 



Having reached the iV''' order in a product, Grassmann equates his 

 result to a scalar and forms successive products by multiplying this 



^ This is precisely wlial Hamilton does. 



' It is easy to prove by the principles of the present paper that n is the same 

 for all sets of units in the space under consideration. 



