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)/Ng (g) over bar (s) of the gluon and quark multiplicities and r((1)) = N'(gg)(s)/N'g (g) over bar (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)).