112 Proceedings of the Royal Irish Academy. 



Thus, the systems V =/|, and cr = -f'Cl are co-reciprocal ; or, 

 when a system compounded from m screws is defined by a linear 

 function (/), the co-reciprocal system is defined by the negative of 

 the conjugate of that function (-/'). 



50. Without changing the origin, it is easy to reduce the function/ 

 by making it depend on r/i co-reciprocal screws. 



To this end ob serve that, if 



fp = - sr„S4p, f'fp = + 5*;/Sr,sr„«/„p. 



The function/'/is obviously self-conjugate ;^ its axes are consequently 

 mutually rectangular : and if they are taken as units, 



f'fi, = %i,,STj,V,,SiJ, = - ^i.STj:, = - i,ST,\ 



This requires generally >S'r„r,, = 0, where u and v are different ; and 

 it is obvious that (Fi, ii) and (T^, 4) are co-reciprocal, because each 

 term of the condition ySTiFj + Si^^ = vanishes. The axes of //being 

 ii'f {-2, &c., its roots are - SV-^, - SV^^, &c. 



Kext, if P is a quadratic in the units, Ti, Tj, &c., are axes, and 

 - >Sri^, - <ST2^ &c., are roots of the new self-conjugate function 



ff'v = :§r,*S'/,24'Sr„p = - :§r,^r,p, 



the units being axes of//. 



It may be remarked that, if (P, ^) is a wrench of the system (P =/^), 

 the (/'P, -fS) is a screw of the reciprocal system, for 



/T =/'.(/!) = -/'(- /O. 



51. Of course a function such as / which generates a quadratic in 

 the units from a linear vector can never be self-conjugate, for its con- 

 jugate produces a linear vector from a quadratic. 



It may, however, be shown to possess a part analogous to the spin- 

 vector of the linear vector functions of Quaternions, and the Theory of 

 Screws affords a convenient approach to this investigation. 



Changing the base-point to the extremity of e, the linear velocity 

 becomes a- = 0-^+ Fifie, and the couple becomes P = P,, + Vo^e, if 

 ctq and Pq are the corresponding values for the old origin. Now, if 



1 Saffp = Sfcfp = Sf'fap. 



