JoLY — The Associative Algebra appliccible to Hyjjerspace. 77 

 of consecutiye pairs of the auxiliary vectors, or more definitely in which. 



This is quite analogous to differentiation for moving axes in three 

 -dimensions. In fact, if in three dimensions moving axes through the 

 origin are instantaneously turning round a direction U6, with an 

 angular velocity TO, a point P, if rigidly connected with them, moves 

 through a distance Yd . OFdt in the element of time cU ; thus, 

 Dct = VO'^dt is the small displacement of the extremity of ■ex. It 

 may be added that, if P is not rigidly connected with the moving 

 axes, but moves relatively to them through a distance dzs, the total 

 displacement is Dw = d'^ + VO'c^ . dt. 



Things are just the same for the curve. Imagine through the 

 ■origin a set of n vectors kept parallel to the varying vectors 

 tti, tta, . . . a„, corresponding to a point moving uniformly with unit 

 velocity along the curve; then, for the rate of space variation of the 

 "extremity of any vector w, 



T^ d^ ^ 

 as 



where — is the velocity of this extremity with respect to the moving 



iixes. In particular, if ra- is fixed relatively to the axes (as are the 

 vectors a), 



^- = 0, and D,zy = Fifiw. 

 as 



7. Analogues of the helix in three dimensions, and of the circle in 

 two, are obtained by supposing the scalars ai, a^, . . . a„-.i to be con- 

 stant, instead of being, as in general, functions of the arc s. 



I shall examine this simple case, and show how the vector equation 

 of a curve may be found when the scalars a are given and constant. 



Using the formula of Art. 5, 



^ The product Q,am consists of a sum of ternary products such as ffiaiaoam, and 

 a sum of linear terms such as amamam+iam = — «»!«»( • omam+i = + amam-ii- The 

 former sum is VzClam ; the latter is ViSlam- 



