74 Proceedings of the Royal Irish Academy. 



The Algeljia considered in the present Paper is that "srhose units 

 ii, {2, . . . in satisfy equations of the type «'/ = - 1, and v<4- «Vs= 0- 



It seems to be due to W. K. Clifford, and the notation in his Papers 

 on " Applications of Grassmann's Extensive Algebra" and " The Classi- 

 fication of Geometric Algebras " is folloTved as closely as conyenient. 

 It is defined to be Associative and Distributive. 



1. A vector, or a right line in ?z-dimensional plane s]5ace regarded 

 as having magnitude and dii'ection, is adequately represented by 

 p = 2v-'s, in Tvhich the x are scalars. 



Any vector coplanar with two given vectors ai and ao is expressible 

 in the f oiTa p = x^a^ + a^ocu. 



Any vector in the same space of thi'ee dimensions as ai, a^, and a^ 

 (which vectors are supposed not to be coplanar) is expressible in the 

 form p = x^ai + 0^202 + x^^a^^ ; and this process may be extended to spaces 

 of higher order. 



The units i^, io, • • • 4 represent unit line vectors mutually rect- 

 angular. Theii' biaary products (//;) represent unit and directed plane 

 areas ; ternary products (v-Ai) represent unit volumes in definite or 

 directed spaces of three dimensions ; and so on for products of greater 

 complexity formed from distract units. 



2. The following discussion of the affections of a curve in n dimen- 

 sions will seiwe not only to illustrate a method, but also to interpret 

 various combinations of symbols in the Algebra under consideration. 

 The method is kinematical, and analogous to that hinted at by Hamilton 

 in his "Elements," and used by Darboux and others with much 

 success : — 



Let p = ^ (s) be the equation of the curve in terms of the length 

 of the arc (s) measui'ed from a suitable x^ouit. The tangent at p is 

 parallel to the unit vector 



Passing from p to a consecutive point on the curve, the tangent changes 

 its direction, and ai becomes 0.1 + D^ui . ds. Put as aj is a unit vector, 

 SaiD^ai = ; so D,ai is at right angles to a^ ; and, writing D^ai = ffiOo, 

 and supposing ao to be a unit vector (the principal normal), a^ is the 

 curvature, or a^ds is the infinitesimal angle between the consecutive 

 tangents. 



Kext, it is easy to establish the equation -D^a^ = a^a^ - rtia^, and to 

 assign geometrical meanings to the new scalar do, and the new unit 

 vector ttg. As Sa.oD,ao = 0, then D,ao will have no component along 03. 



