In this article we present a theoretical method for the study of the general three-body problem by computer simulation developed in the Leningrad State University Astronomical Observatory (LSU AO). This method permits statistical methods to be used for studying the behaviour of triple systems. This is achieved by selecting a representative sample of initial conditions which then reveal general features of the evolution. The main results of numerical experiments on the three-body problem carried out at the LSU AO during the past 25 years have been summarized in the reviews by Anosova (1985), Anosova and Orlov (1985), and Anosova (1986). Systematic studies of about 3 x 10(4) triple systems with negative total energy (E < 0) have yielded the following main results. Most (93.4%) of the systems decay; the decay always occurs after a close triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (4.3%) of stable systems is formed if the angular momentum is non-zero. The qualitative evolution in three-dimensional cases is the same as for planar systems. Small changes in initial conditions sometimes lead to substantial differences in the final outcome. The decay of triple systems is a stochastic process similar to radioactive decay. The estimated mean lifetime is equal to (T) over bar = (107.1 +/- 1.8) crossing times tau for equal-mass components. Thus, for solar mass components and a typical dimension d = 0.01 pc, (T) over bar = (1.6 +/- 1.5) x 10(6) y, and for triple galaxies with M = 10(10)M(0) and d = 50 kpc, (T) over bar = (1.8 +/- 1.7) x 10(11) y. The value (T) over bar decreases with increasing mass dispersion. In this article we also carry out a theoretical analysis of the changes of the integrals of motion in the general three-body problem used as the controls on the calculations. The following basic results have been found: (1) analytical functions of the changes of the integrals of motion during the integration time have been obtained; (2) changes in the integrals of the mass-centre of a triple system do not correlate with the cumulative integration errors; (3) the cumulative changes of the integral of energy are proportional to the sum of squares of the cumulative errors in the coordinates and the velocities of the bodies; (4) the cumulative changes of the square of the total angular momentum are proportional to the product of the square of these cumulative errors. The analysis of the accuracy of computer simulations conducted in LSU AO for the 3 x 10(4) triple systems with E < 0 is summarized by the following basic qualitative results: ( I) the unstable triple systems decay after a mean lifetime <(T)over bar> congruent to 100 tau or (T) over bar congruent to 10(4) (h) over bar (t), where tau is a crossing time, and (h) over bart is a mean integration step. After this integration time (T) the mean cumulative relative changes of the integrals of the energy of the triple systems are equal to (DE) over bar = (0.9 +/- 0.1) x 10(-4) and the mean cumulative relative changes (DL) over bar of the area integrals are equal to (DE) over bar = (1.0 +/- 0.1) x 10(-6); the mean values of the cumulative errors (Dr) over bar, (Dv) over bar in defining the coordinates (r) and velocities (v) of the bodies (during the total integration time (T) over bar) are equal to (Dr) over bar = 0.5 x 10(-3) d, (Dv) over bar = 0.5 x 10(-2)v, where d is the unit of distance, and v is the unit of velocity; the mean local integration errors (of one integration step) are equal to delta r = 5 x 10(-8) d, (delta v) over bar = 5 x 10(-7) v; (2) the process of accumulation of integration errors has a complicated character and correlates strongly with the process of dynamical evolution of the triple systems; (a) because of the strong gravitational interplays of the bodies, the process of the accumulation of the integration errors is very intensive; however, the triple systems with these interplays of the bodies have, as a rule, a small escape time T congruent to 5 tau, and the cumulative calculation errors are small too; (b) in the stable triple systems the local integration errors are practically constant during the numerical study of their evolution, and the calculations can be carried out (if it is necessary) during the time T = (2-3) x 10(3) tau without disturbing the periodical motions of the bodies; (3) thus, in the general three-body problem with different initial conditions, it is not necessary to carry out the computer simulations over long times, as most of the triple systems decay and do not have very long lifetimes; (4) the mean level of the cumulative errors Dr and Dv of the definitions of the coordinates and velocities of bodies in the different triple systems is practically equal.
