q.q = [R (R5- 1(-G4-1k4) + p] [R (R5-1(-G4-1k4) + p].' With the use of properties (4.5) and (4.6) as necessary and rearranging terms, the equations yield u.q = 04 [(n.p) S5 + (b.p) C5] + 74(t.p) d4 (6.36) and

q.q = -2[a4d4 (n.p) a4 (b.p)] S5

 2[a4 (n.p) + C4d4 (b.p)] C5

 274d4 (t.p) + a42 + d42 + p.p, (6.37) where we used the fact that



 n.p nxPx + nypy + nzpz

 R- p= b.p bxPx + bypy + bzpz

 t.p txPx + typy + tzpz



Here again, we note that uz, Pz, u.q, and q.q are linear functions of S5 and C5.

 With the components of u and q and the inner products u.q and q.q computed, a one-dimensional iterative method can be implemented as described earlier with a function f(85) given by


 f(85) = uLqL u.q (6.38)-


which will converge to a value of e5.

..