부산대학교 도서관

Adopting the point of view of Meščersky [Dynamics of a Point withVarying Mass, St. Petersburg, 1897 (in Russian)] that a point of varyingmass represents a body which gains or loses some amount of its mass duringthe process of motion, the differential equations of motion in the problemof two bodies attracting each other according to the law of Newton are setup for the following three astronomical cases. (i) The masses $m$ and $m'$of the two bodies decrease because of radiation (binary stars, the Sun anda comet, under the assumption that the mass lost by the comet leaves itwith zero or almost vanishing relative velocity). (ii) The mass $m$ of oneof the bodies decreases because of radiation or increases because ofabsorption of a dust from a cosmic cloud assumed to be at every moment instatistical equilibrium with respect to the body. The mass $m'$ of thesecond body increases on account of a dust from a cosmic cloud which atevery moment is supposed to be in statistical equilibrium with respect tosome Galilelian system (the Sun and a planet). (iii) The masses of the twobodies increase because of absorption of a dust from a cosmic cloudsupposed to be at every moment in statistical equilibrium with respect tosome Galilelian system (the Sun and a planet, binary stars, under theassumption that the increase of mass of a radiating body by absorption of acosmic dust proceeds more rapidly than the decrease of mass by radiation).An example of a more general case of two bodies with varying masses wasconsidered by Seeliger [Abh. Bayer. Akad. Wiss. München. Kl. II. {\bf 17}$_{2}$, 457--490 (1891)].\parIn case (i) the differential equations of the relative motion have the sameform as in the case of constant mass. In case (ii) the problem can besolved by the method of successive approximations if $m'$ increases veryslowly. In case (iii) the differential equations of the absolute motion(with respect to the above mentioned Galilelian system) possess threeintegrals of momentum. Under a further assumption that the momentumconstants are zero, or that the relative velocity of increase of mass(velocity of increase of mass with respect to itself) for both bodies isthe same, the differential equation of relative motion is$$\frac{d{\bf v}}{dt}+k^2\frac{m+m'}{r^2}\frac{{\bf r}}{r}=\fracd{dt}\log\,\left(\frac 1{m}+\frac 1{m}\right){\bf v},\tag1$$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 thatthe papers by Armellini on the problem of two bodies with varying massesbased on the equation of the form (2) do not describe the real motion in areal 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 areconnected with the coordinates of the real point and the real time by welldefined 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, theauthor proves six theorems on the relative motion of the two bodies withincreasing masses.\parThe rest of the paper deals with (i) the transformation of equation (2) bygeneralizing the transformation of Meščersky [Astr. Nachr. {\bf 132,}129--130 (1893)], which describes the real relative motion in the problemof two bodies with decreasing masses or the motion of an auxiliary point inan auxiliary time in the case of increasing masses, and (ii) withformulations of equivalence between the problem of motion of two bodieswith increasing (decreasing) masses and the problem of motion of a bodywith constant mass under the action of the Newtonian force of attractionand a force of resistance (accelerating tangential force) which may beexpressed as the product of the velocity and some function of time. Thereis an extensive bibliography.