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tokens: our representations to this time span. Test calculations showed that our algorithm can lead to compact and precise representations for both degrees of freedom. With different procedure parameters, the resulting representations contain very different terms. This confirms the flexibility of our algorithm. In order to give representations as physical as possible, we resorted to the knowledge we have for the solution. Following Laskar et al. (2004) , we computed the fundamental frequencies of the secular solar system by applying the NAFF algorithm over time intervals of length 20 Myr for the proper modes , and 50 Myr for the proper modes 4 . From −35 Myr to 5 Myr with a step 0.1 Myr, we generated the samples of the fundamental frequencies ( g 1 , ..., g 8 , s 1 , ..., s 8 ) corresponding to these proper modes. The resulting nominal value of s 5 is about 0.00000015 arcsec yr −1 . The other frequency samples are shown in the panels of Fig. 9 , where the errors are estimated as the difference between the values of a frequency computed from the associated ephemeris and its quasi-periodic approximation. We also show in this figure the Chebyshev approximations of these fundamental frequencies. All of these Chebyshev approximations, the coefficients of which are listed in Table A.2 , were obtained by truncating those of degree 15. The truncation criterion was roughly that the discrepancy in a fundamental frequency should not induce an error in phase larger than 2 π /10 4 over several tens of million years. Fig. 9. Variations of major fundamental frequencies of the solar system over the time interval from −35 Myr to 5 Myr with origin at J2000. These frequencies (in arcsec yr −1 ) are computed by applying the NAFF algorithm to the proper modes of the secular solar system associated with major planets. The errors in the resulting frequency samples are estimated as the difference between the frequencies computed respectively from the original ephemeris and its quasi-periodic approximation. Also shown in this figure are the Chebyshev approximations of these varing fundamental frequencies (red curves), which are specified in Table A.2 . There are two important libration frequencies, r 1 = 0.251085 arcsec yr −1 of the resonance argument and r 2 = 0.117222 arcsec yr −1 of . To include them as additional fundamental frequencies, we expressed the main frequency as The d’Alembert characteristic, , was used to exclude non-physical frequency index vectors. To show to which degree our algorithm is practically useful, we discuss here the following two 100-term representations (R 100 for short), where with f = ( g 1 , ..., g 8 , s 1 , ..., s 8 , r 1 , r 2 ) and k = ( m 1 , ..., m 8 , n 1 , ..., n 8 , k 1 , k 2 ), the phase increment is associated with the main frequency ⟨ k , f ⟩. The way the residuals decrease with increasing number of representation terms in the two representation procedures of R 100 is shown in Fig. 10 . Fig. 10. Convergence property of the