From the integration formulae of Numerov and Subbotin we have developed and programmed for an electronic computer a particular method for integrating the differential equations of cometary motion in special rectangular coordinates, with a variable step and allowing for all planetary perturbations and nongravitational effects over a time interval of 400 yr. Application of this method and our set of programmes to the investigation of the motion of P/Wolf permits us to eliminate the discontinuity that has hitherto existed in the theory on account of the comet's close approach to Jupiter in 1922.

A system is described for processing scientific information with automatic type-setting of tabular text.

The formulae developed by Bielicki for the objective weighting of observations are combined with an orbital improvement method in which the effects of perturbations are included in the differential coefficients. The procedure is applied to the orbit of comet 1953 I.

A new, purely objective criterion is developed for the rejection of observations with nonaccidental errors. A mathematically rigorous procedure is devised for the determination of the weights of individual observations.

An expression is developed for the dependence of the weights of the determination of the elements of the orbit of a one-apparition comet on the number of observations, the length of the observational interval and the apparent motion of the comet during that interval.

The differential correction of nearly parabolic orbits was discussed by the author (Herget, 1939) in the era of lead pencil computing. The Gauss-Marth method is the best one to use whenever the appropriate conditions exist, i.e., |E| < 64° and e nearly unity. The crucial point in the above-cited discussion is the use of the first differences from the Gauss-Marth tables in order to simplify the computation of the partial differential coefficients, namely dB/dA, dC/dA, and dD/dA.

A complex of standard programmes has been developed for computing nearly parabolic orbits. The true anomaly is related to the time using Herrick's formulae. Perturbations are calculated by Encke's method.

The Olbers-Banachiewicz method for computing parabolic orbits has been programmed. A least-square improvement of the orbit is obtained by varying the first and last geocentric distances.

A method for allowing for the effects of nongravitational forces on the motions of comets is summarized. Study of the motions of specific comets indicates that these forces act essentially continuously but have a high inverse dependence on heliocentric distance; there is also evidence for secular changes. The equations of motion employed are discussed in terms of the Whipple icyconglomerate model. Nongravitational parameters are tabulated for all 46 comets observed at three or more perihelion passages. We point out the particular problems that still exist for certain comets and suggest directions for future research.

Some recent investigations on the existence and nature of nongravitational forces acting on comets have required the application of a good deal of high precision numerical procedures. In this paper these procedures are examined critically; special attention is given to cases where difficulties may appear when a comet makes a close approach to a planet or the Sun.

The catalog by Ho Peng Yoke lists ancient Chinese observations of 581 transient astronomical objects of which some 300 might conceivably have been P/Encke, between the years − 156 and + 1600. Using the Gauss-Hill method it was possible to establish roughly the plane of the orbit and the direction of perihelion for P/Encke over this interval of time. All but about 40 of the objects could be eliminated by comparing the observed locations with possible locations of P/Encke. Several more could be eliminated on the basis of their calculated magnitudes and general observability. Large uncertainties in the non-Newtonian motion as yet prevent our certain identification of P/Encke among the remaining possibilities.

Numerical experiments have been made in an attempt to remove the residuals of P/Halley and link the seven apparitions from 1456 to 1910. All efforts to link more than two apparitions using Newtonian equations have invariably failed. However, by the addition of a secular term to the equations of motion, the four apparitions from 1910 back to 1682 can be linked by a numerical integration which represents the observations to contemporary accuracy. When this integration is continued, the apparitions of 1607, 1531, and 1456 show residuals of less than one day in the time of perihelion passage. Prior to 1456 the residuals begin to run off but, with the exception of 1222 and 1066, the apparitions back to 837 show residuals no greater than four days in the time of perihelion passage. The residuals of 20 days in 1222 and 7 days in 1066 appear anomalous but can be made reasonable if the Chinese records are adopted in preference to the European records.

Various sets of osculating elements of P/Brooks 2, derived by Dubyago, are introduced into an N-body integration programme and run from 1686 to 1976. Attempts are made to find a system of elements which links the apparitions before and after the close approach to Jupiter in 1922. The propagation of differential perturbations, and also nongravitational effects, is examined.

Seven apparitions of this comet, from 1925 to 1970, have been linked, with account taken of the perturbations by Venus to Neptune; 202 observations have been represented with mean square error ± 2′.15. The resulting orbit was integrated from 1947 back to 1843 and compared with Möller's results. The result of the comparison after 104 years was found to be very good. The comet's orbital evolution was studied altogether over a period of 400 yr (1660 to 2060). There were nine approaches within 1.5 AU of Jupiter, the comet on one occasion passing through Jupiter's sphere of action. The orbital evolution was free from catastrophic changes, the comet remaining in the Jupiter family throughout the interval. The problem of the comet's discovery was considered.

Orbital elements of P/Giacobini-Zinner have been obtained from 577 observations at the eight apparitions 1900 to 1965 by linking apparitions in pairs. By this technique we established that the comet has a nongravitational secular deceleration amounting, on the average, to 0.081 day/(period)2. As a result of the comet's approach to Jupiter in 1969, a return of the Draconid meteor shower is possible on 1972 October 8d15h45m UT. The 1946 meteor shower was due to meteoroids ejected forward along the orbit of the comet in 1940 with velocities of 14 m s- 1. The meteor showers of 1933 and 1926 were apparently produced by meteoroids ejected in 1900 with velocities of 15.0 m s- 1 forward and 14.5 m s- 1 backward, respectively.

Observations of P/Giacobini-Zinner could not be successfully represented for more than two successive apparitions using Newtonian equations of motion. The comet's motion has a secular deceleration approximately equal to 0.08 day per (period)2. After extending the differential correction to include nongravitational terms in the equations of motion the observations were successfully represented over the time intervals 1900–1946, 1913–1959, and 1939–1965. There is an indication that the magnitude of the nongravitational forces increases with time and that the motion of the comet was discontinuous between 1959 and 1965. A Giacobinid meteor shower prediction has been provided for the 1972 return.

A non-Newtonian orbit has been determined for P/Borrelly which successfully represents all eight apparitions. Although this comet has a substantial secular acceleration in its mean motion, the transverse nongravitational force is time independent over the 64-yr observed interval.

The motion of P/Borrelly has been investigated using all precise published positions and taking into account the perturbations by Venus to Pluto. Nongravitational forces have been found to influence the motion of the comet considerably, formerly as a secular deceleration, more recently as a secular acceleration. By linking the seven apparitions 1904–1960, with allowance for the nongravitational effects, we have obtained a set of elements which represents 802 observations with a mean error of 1′.89.

The orbit of P/Pons-Brooks has been studied using the observations of 1812, 1883–1884 and 1953–1954. A curious deviation of the residuals has been established during the last two months of observation in April and May 1884, but no discontinuity in the orbit due to an impulsive action on the comet has been found. The observations covering two complete revolutions cannot be fitted into one single orbit.

P/Tempel-Tuttle, the comet associated with the Leonid meteor shower, was observed at only two of its last four passages through perihelion, in 1865–1866 and 1965. We have re-reduced the observations in 1865–1866, and with the help of Belyaev's computer programme for numerical integration have linked the two apparitions.