  The charged particle multiplicity in hadronic three-jet events from Z decays
is investigated. The topology dependence of the event multiplicity is found to
be well described by a modified leading logarithmic prediction. A parameter fit
of the prediction to the data yields a measurement of the colour factor ratio
C_A/C_F with the result C_A/C_F = 2.261 +/- 0.014 (stat.) +/- 0.036 (exp.) +/-
0.066 (theo.) in agreement with the SU(3) expectation of QCD. The quark-related
contribution to the event multiplicity is subtracted from the three-jet event
multiplicity resulting in a measurement of the multiplicity of two-gluon
colour-singlet states over a wide energy range. The ratios r=N_{gg}(s)/N_{q
qbar}(s) of the gluon and quark multiplicities and r^{(1)}=N_{gg}'(s)/N_{q
qbar}'(s) of their derivatives are compared with perturbative calculations.
While a good agreement between calculations and data is observed for r^{(1)},
larger deviations are found for r indicating that non-perturbative effects are
more important for r than for r^{(1)}.
