A battery of constant emf \(\mathcal{E}\) and internal resistance \(r\) is connected to a circuit containing two identical resistors, each of resistance \(R\), connected in parallel with each other. In this configuration, a voltmeter connected across the terminals of the battery measures a terminal potential difference \(V_1\). A third identical resistor of resistance \(R\) is then connected in parallel with the first two resistors, causing the voltmeter to measure a new terminal potential difference \(V_2\). What is the internal resistance \(r\) of the battery in terms of \(R\), \(V_1\), and \(V_2\)?