Mills v. Smith was a United States Supreme Court case that dealt with the issue of whether a state could impose a tax on the income of a non-resident. The case was brought by a resident of the state of New York, who argued that the state's tax on his income was unconstitutional. The Supreme Court ruled in favor of the plaintiff, finding that the tax was unconstitutional because it violated the Due Process Clause of the Fourteenth Amendment. The Court held that the tax was an unreasonable burden on the plaintiff's right to travel and to pursue his livelihood, and that it was an arbitrary and capricious exercise of the state's power. The Court also held that the tax was not a legitimate exercise of the state's power to tax, as it was not related to any legitimate public purpose. The Court's decision in Mills v. Smith established the principle that a state cannot impose a tax on the income of a non-resident without a legitimate public purpose.

In Mills v. Smith, the Supreme Court was asked to decide whether a state law that allowed for an appeal of a criminal conviction in certain cases violated the Constitution's prohibition against double jeopardy. The majority opinion held that it did not violate this provision because appeals are part of the same trial and thus do not constitute double jeopardy. Justice Field dissented from this ruling, arguing that allowing an appeal after a criminal conviction constituted two trials and therefore violated the Double Jeopardy Clause. He argued that if states were allowed to permit such appeals, then they would be able to effectively circumvent constitutional protections by simply granting multiple opportunities for review or retrial until they achieved their desired outcome - something he believed was fundamentally unfair and unconstitutional. Furthermore, he argued that permitting such appeals would lead to uncertainty in legal proceedings as defendants could never know when their case had been finally decided due to potential further reviews or retrials down the line.