A partially singularly perturbed linear system of second order ordinarydifferential equations of reaction-diffusion type with givenboundary conditions is considered. The leading terms of first $m$ equations are multiplied by small positive singularperturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first $m$ componentsof the solution exhibit overlapping layers and the remaining $n-m$ components have less-severe overlapping layers. Shishkinpiecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to allthe parameters. Numerical illustrations are presented in support of the theory.