Using (4.4) and (4.5) as needed and expanding yields
u.q = t.(p 1i) R1-1t 12, or
u.q = t.p t.1It.2 which gives
u.q = t.p altx C1 alt y S1 a2 itx C2
2 it S2 d2 Itz. (7.18)
Similarly, the square of the length of vector q, q.q, is given by
q.q = R2-1R1-1 (P 1I)-R1 12] R2-1R1-1 [(p 1)-R1 12 where we factored out R2-1R1-1 in the expression of q from Eq. (7.15). Using (4.5) and (4.6) and expanding again leads to
q.q = p.p + 11.11 + 12.12 2(p.11 + R1- 1p.12 + R1 11.12) or
q.q = -2a2 [(a1 + 1px) C2 + 1py S2] 2d2 1Pz
2a 1px + p.p + a12 + a22 + d22 (7.19)
Equation (7.2) gives rise to a reduced system similar to that of Eqs. ((5.13)-(5.16) with the required shift in indexes,
a3uy S3 + a3ux C3 + a4o5 S5 4o5d4 C5 = r1 (7.20)
U3ux S3 U3uy C3 + G4a5 C5 = r2 (7.21)
..