tokens: the top panel of Fig. 2 , we show the case in which the apparent periodical variability of the spin period of LMC X-4 is tentatively described with the sinusoidal function: Here, is the mean value of the source spin period over the whole observational period (in seconds), P 3 is the characteristic time-scale of the variability, T 0 is the time of phase zero, and A the amplitude of the variability ( t , T 0 , and P 3 are all in units of days). Figure 2. Tentative description of the pulse period variations in LMC X-4 with two mathematical functions: the sinusoidal one described by equa-tion ( 1 ) (top panel) and the linear one described by equation ( 2 ) (the bottom panel). See the text for details. The equation given above describes the data reasonably well, and we obtained P 3 ≃ 5090, A = 5.3 × 10 −4 s, and s. Only the first two points obtained from the SAS observatory deviate from the qualitative periodical function. It is interesting to note that only these measurements together with the one based on the HEAO-2 data were obtained during episodes of the source flaring activity. All other spin period values were measured during quieter periods in the high superorbital phase. It is thus possible that some mechanism is at work during the flares to produce changes in the observed source pulsation period. As the available observational data (Fig. 1 ) are rather sparse, we cannot exclude that the different spin-up and spin-down episodes alternate in a more complicated fashion than a simple sinusoidal trend. Therefore, we show in the bottom panel of Fig. 2 an another example in which data are described by a sequence of power laws with identical absolute values of the slope but alternated signs: In the equation above, is the period derivative, T i is the time interval preceding when the pulse-period derivative changes sign, and P i is the pulse period in the time interval T i (in the equation all time intervals are expressed in days and all spin period values in seconds). According to this model, the characteristic rate of the pulse period change would be s −1 . 4 DISCUSSION The results presented in the previous section show that LMC X-4 undergoes cyclical transitions between spin-up and spin-down phases, with torque reversals occurring roughly every ∼6.8 yr. The spin period variability could be reasonably well described by using either a sinusoidal function, which would suggest the presence of smoothly periodical variations between two values of maximum and minimum spin periods, or a slightly more complicated function, relaxing the needs of assuming a strictly periodical phenomenon. At present, given the irregular and large spacing among all data available, it is not possible to firmly distinguish among these two possibilities. Nevertheless, below we discuss briefly several possible mechanisms that could give rise to quasi-periodical torque reversals in systems like LMC X-4. Third body. Concerning the strictly periodical case (i.e. equa-tion 1 ),
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