16 Proceedings of the Royal Irish Academy. 



scalar by new points. At this place the associative law breaks down. 

 We can imitate Grrassmann's procedure not by ear-marhing the function 



PiP^Pz ••■Vm 



but by dividing the function F^-sri-irs . . . -zr^ by O, and proceeding 

 along the associative lines of our algebra. Of course the laws of the 

 algebra afford the equation 



f22 ^ (^_ 2^A^+(7V-l) + (yV-2) ^ ^_ \\\N{N^\-)^ 



We have the Grassmann product 



Pi Pi ■■• PnPn^I ■••PN+m., 



and our equivalent 



We have moreover the conception of complements expressed by the 

 functions 



F„-zri-z<ro . . . TxTn and ^iv-»'z^»+i'5E^H+2 • • • "^n • ^~^' 



We have still to see how we can include in the associative algebra 

 propositions like the following : — " The product of two posited quanti- 

 ties which have no common figure is some multiple of the connecting 

 figure," or "The product of the two posited quantities which must 

 have a common figure is the common figure multiplied by a number." 



Take these two functions having a common figure 



T^i-ZtTi-SETo . . . -ZET,, and VviPlpI • • • Pm> 



and, by the process of (6.), reduce them to the form 



Vn'ni^i^i • • • e«o-iO-2 . . . (Tn.t aud K.yjj^e^e^ , . . etT^Tz . . . t,„.„ 



where 7?i is the point symbol of some point in the common figui'e, and 

 ejea, &c. are vectors of the common figure. For these we may write 



VtVl^l^Z •••€«. Vn-l(y\0-1 • ■ ■ Or.n-t aud Vtr]i€.^^ . . • C^ . F.^.^TiTs . . . T„,-t ', 



since the symbols F"„ and V„^ are superfluous, o- and t being exclusive 

 of yj and €, or at right angles to them. The complete product of these 

 is 



± Vty]x^2^z ... ^f VtVi^z^ • • • ^i- K-tO-iO-2 • . . a;^t • K^tTiT. . . . r„_«, 



or if we attend particularly to the sign, though this is immaterial, we 



