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This process is a t once much simpler than that described by Kopal, much less time consuming, and more convenient for machine computation. Moreover, it is based on a firm mathematical foundation. 4). For the n-body problem the matrix (uJ is not difficult to derive. 4) are also required to obtain the partial derivatives of the vehicle's position and velocity with respect to initial parameters. This use of the variational system for this purpose will be described in Section 6. 1). 1) can also be extended to a consideration of round-off errors following the method outlined in Section 3 but this will not be pursued here.
Thus, the major error is accounted for by a time discrepancy so that the integral is actually a t a point on the orbit but a t a different time. Thus even though the integral is constant to N significant digits the actual position and velocity may be considerably worse. The energy integral does provide a good negative test-the method is generally poor if the integral does not remain constant. In some cases the energy integral has remained constant to seven digits, while the position coordinates were correct to only two digits.
For groups which are heavily engaged in the problems of orbit prediction, it is recommended that a t least two distinct methods be available for orbit integration. 6 Estimates of Accumulated Truncation and Round-off Errors Theories have been developed which make it possible to obtain estimates of the accumulated truncation and the accumulated round-off error a t a given time. If these estimates were available then it would be theoretically possible to choose an integrating step which would minimize the sum of 39 SATELLITE ORBIT TRAJECTORIES these errors and a t the same time an indication of the over-all accuracy would be available.
Advances in Computers, Vol. 3