464 THE THEORY OF SCREWS. [419,
419. On the Intervene through which each Object is Conveyed.
Given an object, Xlt X2, X3, Xlt find the intervene through which it is
conveyed by the transformation, when
4" = 0
is the infinite quadric.
In this equation substitute for Xj + 6Ylt &c.; and, remembering that
Y, = p1X1, &ze., we have,
(Xi + pjöJTj) (X., + p20X2) + X + p:,dX:i) (X4 + p40X4) = 0,
or 0- (pjpsZjXa + Xp3p4X3 Xi)
+ $ (pi-^i^2 + p2X4X2 + Xp3X3Xi + XpiX3Xi)
+ X,Xa + XJTg-Xa = 0.
We simplify this by introducing
P1P2 ~ psPi>
and writing Å.X3X4 4- XiX2 = <f>, whence the eq nation becomes
(FpiPz (1 + ’M + $ lPi + P'2 + </> (p3 + Pj)] + (1 +</>) = 0 ;
hence if 8 be the intervene, we have,
„„„ X - 1 Pi + + </> (ps + pt) _
2Vpjp2 1 + 0
or, if we restore its value to </>,
, o _ 1 -XpXs (pi + pa) + (øs + pi) XX3Xi..
i^pTp, XiX. + xX.Xi
If Pl + p2 = Pi + Pi,
then cos 8 = ;
ZvpiPt
i.e. all objects are translated through equal intervenes. This is the case
which we shall subsequently consider under the title of the vector, as this
remarkable conception of Clifford’s is called. In this case, as
P1 + P-2= Pi+ Pi,
and also, p^ = p3p4,
we must have p, = p3, and p2 = p4,
or Pi=pt, and p2 = p3.
In either case the equation for p will become a perfect square.