The temperature \(T\) at time \(t\) and at a distance \(r\) from the center is connected by means of the partial differential equations \[ \frac{\partial T}{\partial r} = k \left(\frac{\partial^2 T}{\partial r^2} + \frac2r \frac{\partial T}{\partial r}\right) + \sigma \frac{\partial}{\partial t} \left(3\Psi + r\frac{\partial \Psi}{\partial r}\right) + \Phi, \] \[ 3r \frac{\partial^2\Psi}{\partial r^2} + 12 \frac{\partial \Psi}{\partial r} = 5 \delta \frac{\partial T}{\partial r}, \] with quantities \(\Phi(r, t)\), \(\Psi(r, t)\) such that \(\rho C_v \Phi(r, t)\) is the heat generated per unit time per unit volume by the radioactive matter and \(r\Psi\) is the increment of the radius vector which initially had the value \(r\). In these equations \(C_v\) and \(C_p\) are the specific heats at constant volume and constant pressure, \(\rho\) is the density, \(\delta\) the coefficient of thermal expansion and \(\sigma\) a coefficient defined by the equation \(3\sigma\delta C_v = C_p - C_v\). The supplementary conditions are that as \(t\to 0\), \(T\to f(r)\), \(\Psi\to 0\) and that as \(r\to R\), \(T\to 0\) and \(5 \Psi + 3r\frac{\partial \Psi}{\partial r}\to 0\). The problem is solved by means of the Laplace transformation, a biorthogonal set of functions and the multiplication theorem for integrals of Laplace's type.