We study the connection between the Hilbert series, \(Hilb_ R(t)=\sum^{\infty}_{0}\dim_ k((\bar X_ 1,\bar X_ 2,\bar X_ 3)^ i/(\bar X_ 1,\bar X_ 2,\bar X_ 3)^{i+1})t^ i,\) and the Poincaré series, \(P_ R(t)=\sum^{\infty}_{0}\dim_ k(Tor^ R_ i(k,k))t^ i=\sum^{\infty}_{0}\dim_ k(Ext^ i_ R(k,k))t^ i\) for rings of the type \(R=k[X_ 1,X_ 2,X_ 3]/(f_ 1,...,f_ r)\) where the \(f_ i's\) are forms of degree two and k is a field. It is well known that the Hilbert series are always rational, and we show that the Poincaré series are determined by the Hilbert series and that they are rational. As a corollary we can conclude that local rings (R,m) of embedding dimension three and with \(m^ 3=0\) have rational Poincaré series. Furthermore we show that graded Artinian rings of length \(\leq 7\) (and thus graded Cohen-Macaulay rings of multiplicity \(\leq 7)\) have rational Poincaré series. - It is known that there are local rings of embedding dimension five with \(m^ 3=0\) and graded Artinian rings of length 11 with transcendental Poincaré series; see D. J. Anick, Ann. Math., II. Ser. 115, 1-33 (1982; Zbl 0454.55004).