JoLY — On the Place of the Aiisdehmmgslehre. 



17 



must replace ± by the definite value (- 1)'^" *\ 

 product, if we select the part 



Frora this complete 



KVi^i^s • 



V . 



-tVl^2 



e(0-iO-2 



0-n-tTir2 



we have the Grassmann "product" when we divide by O, the 

 product of. a set oi n + m - t units in the space containing^both figures. 

 The symbols F„_j and V^^t are suppressed, being unnecessary. 



"When the functions have no common figure, the Grassmann 

 product is simply 



^^■m+ii'^l'^2 • • • '^uPlP2 • • • Pm- 



As examples, take two-point line vectors V^-m-i^z ^^^ T^^PiPz- I£ 

 they are contained in a common plane, since they have a common 

 point, they may be replaced by ViWo-^'i and Fs-ziro-zr'a, and the Grass- 

 mann product is -ro-g FsiTo-zr'i-iEr'a divided by the product of the three units 

 when the product is planimetric. If they have no common point, the 

 "product" is F^-zri-rrajOipa, divided by the product of the four units 

 when the product is stereometric. The principles of the associative 

 algebra allow us to write 



Vi-UTi-sr^pipo = Vi F2W1W0 F2P1P2 = F4 V2P1P0 V^TJTiTJfz 



by the law of interchanges or otherwise. Hence we see that "mul- 

 tiplication" of two lines in space is a " commutative operation." 



It would be tedious to dwell on other parallelisms between the 

 AusdelinungsUhre and the exceedingly restricted use of the general 

 associative algebra to which it corresponds. I may mention, however, 

 that Grassmann's quotients or matrices are simply operators of a very 

 special kind of the type q { ) q~'^ considered in a paper on " The 

 Associative Algebra of Hyper-space."^ Also the continued Grassmann 

 product F„-zzri^2 • • • T^n niay be expressed in the form of a deter- 

 minant 



■Wi, taT^, . . . -nr„, 



■Wi, 



consisting of the same row of vectors repeated n times, if we agree 

 that the determinant shall be expanded just as if its constituents were 



1 Proc. R. I. A., 3rd Ser., Vol. V., No. 1. 

 K.I. A. PEOC, SEK. III., VOL. VI. 



