56 



PROCEEDINGS OF SECTION A. 



It is obvious that a change in the origin can make no difference 

 in the direction of a screw reciprocal to five given screws ; and conse- 

 quently the expression (vii.) must remain unaltered if for /a,, /a,, &c., 

 there be substituted (/x,— VpA.,), (/a,— VpA.,), &c , and this must be 

 true whatever p may be. The verification is made by first noting that 

 this change does not affect w„ (o„ &c., Wj, as was indeed to be expected, 

 because these are the vectors of the axes of the reciprocal cylindroids. 

 It is therefore only necessary to make the changes in /x, , . p.,, &c., in 

 (x.) leaving w^ . . w^ unaltered, and it is easy to verify by well-known 

 quaternion formulse that p disappears when the terms are summed. 



As a further verification of the expressions of A. and p. we shall take 

 the case of six canonical co reciprocal screws* lying two by two on three 

 intersecting rectangular axes. If the axes be i,j, /c, and the pitches 

 be ± a, ik b, =b o respectively, we shall take the origin at the inter- 

 section and thus have A, = i, p., = ai; A, := t, p., =: — ai; A^ = ;", p.3 = hj : 



K = j, /^4 = — hJ ; K = k, /x^ = ck. 



In each case YX,'fx^ = 0; thus showing that all the screws pass througk 

 the origin ; and for the pitch we have SA,/p, = a. 



Substituting these values in the expressions (vii.) and (viii.), we 

 obtain — 



A := 1 ahk p, = — 4 ahck ; 



thus VA/p, := and S p,/A = — c ; showing that the reciprocal screw is, as 

 it ought to be, the sixth screw of the set of canonical coreciprocals. 



A 4-system of the most general type may be defined by the four 

 screws — 



• — ai, i ; — bj,j : cA', I' ; — ck, k ; 



and the general screw of the system will be represented by — 

 p,'= axi-\-byj-]- zk 

 A'= — xi— yj-\-z^k ; 



when X, y, z, z" are any scalars. This is verified by observing that p,, A 

 is reciprocal to both the screws ai, i and hj, j, as are also the four 

 original screws. 



If we substitute in formulae (vii.) and (viii.) as follows: — 

 C p.,= —ai, f p^„=: — bj \ ix^^=ck i fx^=- —ck \ p.,= axi -t hyj + zk 

 |A,= i \K= j l'^3 = ^ (^4== ^ » \=;— ~^^ ~yj +^'^> 



it is easy to verify that all the terms in p, and all the terms in A vanish 

 identically. Thus we obtain the folIoAving theorem : — 



If the five screws (/a, A,) . . . (p.^, A-) belong to a 4-sysrem (or a 

 portion to any system of lower order), then the expressions for A and p. 

 given in formulae (vii.) and (viii.) vanish identically. 



* See Sir Robert Bairs Treatise on the Tlieory of Screws, Cambridge, 1900, p. 38. 



