부산대학교 도서관

Adopting the point of view of Meščersky [Dynamics of a Point with Varying Mass, St. Petersburg, 1897 (in Russian)] that a point of varying mass represents a body which gains or loses some amount of its mass during the process of motion, the differential equations of motion in the problem of two bodies attracting each other according to the law of Newton are set up for the following three astronomical cases. (i) The masses $m$ and $m'$ of the two bodies decrease because of radiation (binary stars, the Sun and a comet, under the assumption that the mass lost by the comet leaves it with zero or almost vanishing relative velocity). (ii) The mass $m$ of one of the bodies decreases because of radiation or increases because of absorption of a dust from a cosmic cloud assumed to be at every moment in statistical equilibrium with respect to the body. The mass $m'$ of the second body increases on account of a dust from a cosmic cloud which at every moment is supposed to be in statistical equilibrium with respect to some Galilelian system (the Sun and a planet). (iii) The masses of the two bodies increase because of absorption of a dust from a cosmic cloud supposed to be at every moment in statistical equilibrium with respect to some Galilelian system (the Sun and a planet, binary stars, under the assumption that the increase of mass of a radiating body by absorption of a cosmic dust proceeds more rapidly than the decrease of mass by radiation). An example of a more general case of two bodies with varying masses was considered by Seeliger [Abh. Bayer. Akad. Wiss. München. Kl. II. {\bf 17}$_{2}$, 457--490 (1891)]. \par In case (i) the differential equations of the relative motion have the same form as in the case of constant mass. In case (ii) the problem can be solved by the method of successive approximations if $m'$ increases very slowly. In case (iii) the differential equations of the absolute motion (with respect to the above mentioned Galilelian system) possess three integrals of momentum. Under a further assumption that the momentum constants are zero, or that the relative velocity of increase of mass (velocity of increase of mass with respect to itself) for both bodies is the same, the differential equation of relative motion is $$ \frac{d{\bf v}}{dt}+k^2\frac{m+m'}{r^2}\frac{{\bf r}}{r}=\frac d{dt}\log\,\left(\frac 1{m}+\frac 1{m}\right){\bf v}, \tag1 $$ which by means of a suitable transformation is reduced to the form $$ d^2\boldsymbol\varrho/d\tau^2+\mu\boldsymbol\varrho/\rho^3=0, \tag2 $$ where $\mu$ is an increasing function of $\tau$. From this it follows that the papers by Armellini on the problem of two bodies with varying masses based on the equation of the form (2) do not describe the real motion in a real time, but the motion of some auxiliary point in an auxiliary time. This fact, however, does not diminish the value of Armellini's work, because the coordinates of the auxiliary point and the auxiliary time are connected with the coordinates of the real point and the real time by well defined formulas. Starting with the equation (1) and making no reference to (2) or to results known from it, for example, by the work of Armellini, the author proves six theorems on the relative motion of the two bodies with increasing masses. \par The rest of the paper deals with (i) the transformation of equation (2) by generalizing the transformation of Meščersky [Astr. Nachr. {\bf 132,} 129--130 (1893)], which describes the real relative motion in the problem of two bodies with decreasing masses or the motion of an auxiliary point in an auxiliary time in the case of increasing masses, and (ii) with formulations of equivalence between the problem of motion of two bodies with increasing (decreasing) masses and the problem of motion of a body with constant mass under the action of the Newtonian force of attraction and a force of resistance (accelerating tangential force) which may be expressed as the product of the velocity and some function of time. There is an extensive bibliography.