These results coincide entirely with those which would have been obtained by viewing S$_{D}$ as a single body. Counting the number of degrees of free- dom of S$_{D}$ by consideration of the individual particles is something which had never been done before. Just the three-dimensional case proved to be complicated enough. Even for Albert Einstein, who was never able to write the correct expressions for the number of de- grees of freedom of S$_{3}$ in [5], in spite of several revisions he made of this book. as rigid bodies. Thus, a diatomic molecule has five degrees of freedom and a polyatomic molecule has six. According to the Equipar- tition of Energy Theorem, each of these de- grees of freedom is associated to an energy of quantity 1 $_{2}$kT . Hence, the internal energy U of a diatomic molecule is U = 5 $_{2}$kT and that of a polyatomic molecule is U = 3 kT . Multi- plying these results by Avogadro´s number, N$_{A}$ = 6 × 10 $^{23}$, gives the internal energy of an ideal gas, which is U = 5 $_{2}$N$_{A}$ kT = 5 $_{2}$RT and U = 3 N$_{A}$kT = 3 RT for diatomic and polyatomic gases, respectively [6],[7]. There seemed to be contradictions be- tween the values of N$_{3}$ obtained viewing S$_{3}$ as a sole body and those reached by considering the individual particles. This was only be- cause the count of the degrees of freedom of S$_{3}$ from the latter standpoint was never done properly. In this paper, we prove that both methods are equivalent, not only in three, but in any number D of dimensions. The heat capacity at constant volume C$_{v}$ is related to the internal energy by the expres- sion C$_{v}$ = ( ∂U $_{∂T}$) , thus C$_{v}$ = 5 $_{2}$R for diatomic gases and C$_{v}$ = 3 R for the polyatomic ones. The heat capacity at constant pressure C$_{p}$ is given by C$_{p}$ = C$_{v}$ + R . This may be of interest for those who study the Kinetic Theory of Gases. In the Kinetic Theory of Gases and more specifi- cally, in the Ideal Gas Model, the internal energy and the heat capacities at constant volume and constant pressure of an ideal gas are calculated as functions of the degrees of freedom of the gas, which are counted per molecule. And for molecules consisting of more than one atom, the number of degrees of freedom is calculated treating the molecules The values of the heat capacities predicted using the Ideal Gas Model agree very well with the values obtained experimentally in the case of diatomic gases, but fall rather short for polyatomic gases [6], [7]. This is due to the fact that besides the energies as- sociated with the traslational and rotational degrees of freedom, there is also vibrational energy. This vibrational energy is quanti- cized, which means that it does not spread over a continuous spectrum of values, but is distributed in discrete states [7], [8]. In the case of most diatomic molecules, 6