Using the helicity formalism, we calculate the combined angular distribution function of the two gamma photons (\(\gamma _1\) and \(\gamma _2\)) and the electron (\(e^-\)) in the triple cascade process \(\bar{p}p\rightarrow {}^3D_3\rightarrow {}^3P_2+\gamma _1\rightarrow (\psi +\gamma _2) +\gamma _1 \rightarrow (e^- + e^+) +\gamma _2 +\gamma _1\), when \(\bar{p}\) and \(p\) are arbitrarily polarized. We also derive six different partially integrated angular distribution functions which give the angular distributions of one or two particles in the final state. Our results show that by measuring the two-particle angular distribution of \(\gamma _1\) and \(\gamma _2\) and that of \(\gamma _2\) and \(e^-\), one can determine the relative magnitudes as well as the relative phases of all the helicity amplitudes in the two charmonium radiative transitions \({}^3D_3\rightarrow {}^3P_2+\gamma _1\) and \(^3P_2\rightarrow \psi +\gamma _2\).