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(abbreviated) In this note we consider, in a weak-field limit, a relativisticlinear motion of two particles with opposite signs of masses having a smalldifference between their absolute values $m_{1,2}=\pm (\mu\pm \Delta \mu) $,$\mu > 0$, $|\Delta \mu | \ll \mu$ and a small difference between theirvelocities. Assuming that the weak-field limit holds and the dynamical systemis conservative an elementary treatment of the problem based on the laws ofenergy and momentum conservation shows that the system can be acceleratedindefinitely, or attain very large asymptotic values of the Lorentz factor$\gamma$. The system experiences indefinite acceleration when itsenergy-momentum vector is null and the mass difference $\Delta \mu \le 0$. Whenmodulus of the square of the norm of the energy-momentum vector, $|N^2|$, issufficiently small the system can be accelerated to very large $\gamma \propto|N^2|^{-1}$. It is stressed that when only leading terms in the ratio of acharacteristic gravitational radius to the distance between the particles areretained our elementary analysis leads to equations of motion equivalent tothose derived from relativistic weak-field equations of motion of Havas andGoldberg 1962. Thus, in the weak-field approximation, it is possible to bringthe system to the state with extremely high values of $\gamma$. The positiveenergy carried by the particle with positive mass may be conveyed to otherphysical bodies say, by intercepting this particle with a target. Suppose thatthere is a process of production of such pairs and the particles with positivemass are intercepted while the negative mass particles are expelled from theregion of space occupied by physical bodies of interest. This scheme couldprovide a persistent transfer of positive energy to the bodies, which may beclassified as a 'Perpetuum Motion of Third Kind'.

On relativistic motion of a pair of particles having opposite signs of masses