For any p>1 and for any sequence $\\{ a_j \\}^\\infty_{j=1}$ of nonnegative numbers, a classical inequality of Hardy gives that$$\\sum^n_{k=1} \\left({\\sum \ olimits^k_{i=1} a_i\\over k}\\right)^p\\les \\left({p\\over p-1}\\right)^p\\sum^n_{k=1}a^p_k \\quad {\\hbox{for each}}\\;n \\; \\in \\; {\\open {N}},$$unless all $a_j=0$, where the constant $[p/(p-1)]^p$ is best possible. Here, we investigate this inequality in the case p=2, and show how it can be interpreted in terms of symmetric ultrametric matrices. From this, a generalization of Hardy's inequality, in the case p=2, is derived.