JoLY — The Associative Algebra applicable to Hyperspaee. 1 15 



54. Before leaving the Theory of Screws, which has been both 

 instructive and suggestive in the study of this Associative Algebra, 

 I shall say a few words on the canonical representation of a screw in 

 hyperspace. By Art. 51, the couple at the extremity of the vector e, 

 arising from a couple Tq, and a force ^ at the origin as base point, is 

 r = ro+ Fjfe. Multiplying this by ^~'^, and separating the parts of 

 the product of the first and third order in the units, two equations 

 are obtained, 



Fl^T = Fi^-To + V,t' r,ie, and V.t'T = V,t'T„ 



of which the first contains c, but the second is independent of it. 

 Now, it is easy to see that e may be chosen so that 



r.t^T = (or r,$T = 0); 



and in fact, as F^^"^ V^^e = e - iS^"^£, 



the condition is satisfied, provided c lies on the right line, 



e = - Fll-To + xt 



This line is the axis of the screw. 



If Pq = /^, so that the wrench belongs to the system /, 



may be regarded as the equation of the assemblage of the axes of 

 wrenches of the system /, if ^ is allowed to vary arbitrarily. If, 

 iowever, | is constrained to remain parallel to a plane, or if 



^ nil + t^2. 



where li and I2 are fixed ; but Hs a varying scalar, 



or € = - Fi (ii + t^.X' (ri + ^r,) ^x{ti-^ ti,) 



is the equation of the locus of the axes of wrenches compounded of 

 two given wrenches ; and this locus is the analogue of the cylindroid. 

 Similarly, the equations of the assemblages of the axes of wrenches 

 compounded of any number of given wrenches may be vrritten down. 

 In any of these equations, on putting x = 0, the equation of the locus 

 of feet of perpendiculars from the arbitrary origin on the axes is 

 obtained. 



In the next place, the function ( Fg^'T) of the third order which 

 is invariantal with respect to a change of base-point, is the analogue 

 of the pitch. It is easy to justify this apparent anomaly, for the 

 effect of the wrench is not confined to any one definite space of three 



