JoLY — The Associative Algebra applicable to Hyperspace. Ill 



48. The Theory of Screws in an w-space furnishes another linear 

 function of some interest. • 



Let Ti, Fs . . . r„ represent couples (quadratic functions of the 

 units), and $i, $2 • • • ^m corresponding forces, referred to the origin 

 as base-point ; the wrenches determined by these quantities are sup- 

 posed to be contained in an m-space, and to be independent one of 

 another. 



If ti, ti . . . t„i are scalars, V = %tj^i, and ^ = %t-^^x are general 

 expressions for the couple and force of any wrench compounded from 

 the given wrenches (Fi, ^1), (Fa, ^2)? &c. 'J'hen it is obvious that the 

 scalars (t) may be chosen, so that a new system of wrenches (F'l, ^'1), 

 (F'2, 1'j)) &c., is obtained in which the forces are all mutually perpen- 

 dicular; or, dropping the accents, the system depending on the m 

 wrenches may be defined by the m new wrenches 



(Fi,0(F2,4), . . .(r„.,4), 



where «i, 4 • • • im are a set of unit vectors in the m-space. 



ISTow, let fp = - SFi Si-^p, and it is evident that fi^ = Fj, fu = T^, &c. , 

 and generally that f^t-yii = %tiTi ; or, if (F, ^) is any wrench of the 

 system, it is obvious that F and $ are connected by the relation F -/$. 



The function / defines the system of wrenches compounded from 

 m independent wrenches in the m-space, so that this system may be 

 designated by the single symbol/. 



49. Again, referred to the origin as base-point, let Oi, O3 . . . Q,„ 

 be the angular velocities (quadratic in the units), and o-i, 0-2 ... o-^ 

 the corresponding linear velocities of any ?n independent motions in 

 the m-space. 



The screw (cr, O) is co-reciprocal to (F, ^), if ^'FO + Sia- = 0;^ and 

 it is co-reciprocal to the system F =/$, if 



Sfi^ + Sia- = 0, or if S^ (/'O + 0-) = 0, 



where /' is the conjugate of /.^ In other words, the linear vector 

 /'O + o- must be perpendicular to cr. 



If then the system of screws (S^io-, S^iOj) is co-reciprocal to the 

 system F =fi, it is necessary that the screws should satisfy /'O + o- = 0, 

 for otherwise in the given w^-space m independent vectors would be 

 perpendicular to cr. 



1 See Arts. 45, 46 in justification of the expression — Sm for the yf^oik done by 

 a couple. 



2/' is defined by SPff) = Spf'P, where P is quadratic in the units. In full, if 

 fp = - ^TiShp, f'P = - SJi'SriP. 



