Uall — Contributions to the Theory of Screws. 65 



pitch-qiiachic whicli is parallel to A. Hence (>, the vector lo any point on the 

 diameter, is given by the equation 



p = h(V,p\. X-' - V<t,'\ . A-) I .'A, (103) 



where x is a variable scalar. 



It is easy to see that this may be written in the more concise fonn 



p = i^'' i<t>'>^ - '/'^) + ^^^• 



Multiplying by A and taking the scalars, and supposing A to be a unit vector, 



we have x = - Sp\. 



If therefore i,j, Jc have their usual siguification as any three unit-vectors 



which are mutually rectangular, and if p be the vector to the centre of the 



pitch-quadric, then 



p = ^iifi- ijt'i) - iSpi, 



p = hj{i>j-i>'j) -y%'. 



p = ^K {<pJc — (f> Jc) — kSplv. 



Adding these three equations, and remembering the well-known quaternion 



formula 



p = - iSpi - jSpj — kSph, 

 we have 



2p = i (i0z +y^y + ^(^k) - -i {iip'i +j<p'j + kij)'!;:) 



= |- V{:ifi +j(t>j + krpk) - i V(:i<p'i +j<p'j + k,p'k). (104) 



We now make the following characteristic transformation, derived, of 

 course, from the wonderful manipulations of his symbols introduced by 

 Hamilton : — 



- i V(i'(l>i +j(p'j + kf'k) 



= h ViWi + ^H'J +M'^) = h V{l>'i Vjk + ^'j Vki + ^'k Vij) 

 = IkSj^'i - ySk^'i + ^iSk,t,'j - likSi<p'j + ySif'k - iiSj<t,'k 

 = I kSiipj - ySitpk + J iSj(j)k - ^ kSjcpi + ySkfi - i iSk<ltj 

 = l(jSk<{,i - kSjipi) + i(kSi<pj-iSk<pi) + h{iSj(f>k -jSi(t,k) 

 = i Vjk<pi + i Fki(j)j + h Vij<pk = ^ V{i(pi +j<j>J + k<j>k). 



Hence we obtain from (104) the result* 



p = ^ V{i<l>i +j<t,j + l><pk). (105) 



This shows how the vector from the origin to the centre of the pitch-quadric, 

 or rather of the system of ^-pitch-quadiics, is expressed in terms of the 

 linear vector function. 



It is easily seen that if \, fi, v be any other set of unit vectors at right 



angles, 



jS(i^i +j<j)j + k(pk) = S{\(pX + p(pn + v>j,v). (106) 



• Joly, " Manual," pp. 97-159. 



[9*] 



