In this contribution we present two new proposals for measuring the generalrelativistic gravitomagnetic component of the gravitational field of the Earth.One proposal consists of the measurement of the difference of the rates of theperigee $\psi$ from the analysis of the laser--ranged data of two identicalEarth'artificial satellites placed in equal orbits with supplementaryinclinations. In this way the impact of the aliasing classical secularprecessions due to the even zonal harmonics of the geopotential would becanceled out, although the non--gravitational perturbations, to which theperigees of LAGEOS--type satellites are particularly sensitive, should be alimiting factor in the obtainable accuracy. With a suitable choice of theinclinations of the orbital planes it would be possible to reduce the periodsof such insidious perturbations so to use not too long observational timespans. However, the use of a pair of drag--free satellites would greatly reducethis problem, provided that the time span of the data analysis does not excessthe lifetime of the drag--free apparatus. In the other proposal the differenceof the rotational periods of two counter-revolving particles placed on afriction-free plane in a vacuum chamber at the South Pole should be measured inorder to extract the relativistic gravitomagnetic signal. Among other verychallenging practical implications, the Earth's angular velocity$\omega_{\oplus}$ should be known at a $10^{-15}$ rad s$^{-1}$ level from VLBIand the friction force of the plane should be less than $2\times 10^{-9}$ dyne.
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