[ 69 ] 



IX. 



SOME NEW RELATIONS IN THE THEORY OF SCREWS. 



By PROFESSOR C. J. JOLT, M.A., D.Sc, F.T.C.D. 



Eead December 14, 1903. Published January 26, 1904. 



In a paper, communicated to the Academy two years ago, and since 

 publislied in the Transactions,^ I wrote : — 



" As another example, if /a, A represent a wrench, A being the 

 force, and /x the couple at the origin as base-point ; the ratios of the 

 independent terms of the array 



(Ai, A2, . . . A,J 

 include all the invariants of an w-system of screws." 



In other words, the ratios of independent terms are the same for 

 all screws of the system. 



Naturally one seeks to reduce invariants to their simplest form, 

 but it is possible to overlook some important relations involved in the 

 crude and unreduced expressions. Consequently I did not notice the 

 following theorem until recently, in framing an example for a text- 

 book. It is one of many, but I have not leisure at present to examine 

 the subject in detail. 



By the laws of quaternion arrays, two independent terms in the 

 array for a three-system of screws afford the invariant 



S/AiA2A3 + S/^2A3Ai + S/X3AiA2 J 



■ ■ ^^ ill ~7~ ~r 0» 



SA1A2A3 



wliere a, b, and c are the pitches of the principal screws of the system. 

 In this replace /x^ hj piXi + YaAj, &c., where ^1 is the pitch and a the 



^ Vol. xxxii, Section A, p. 30. See also p. 28. 



R.I. A. PROC, VOL. XXIV., SEC. A.] K 



