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Lonberg and Meyer [Phys. Rev. Lett. 55, 718 (1985}]have recently discovered in polymer nematics the existence of a critical value r, =0.303 for the ratio r =%2/Kl between the twist and splay Frank elastic constants, below which a periodic splay-twist distortion is energetically favored with respect to the weB-known aperiodic splay distortion which appears in a Freedericksz transition. In this paper the effect of weak-anchoring conditions on the critical r value and on the other parameters of interest is theoretically investigated. Analytical expressions between these parameters are given. It is shown that the periodic transition is not allo~ed in the range 0.5 g r g2. Outside this range either the periodic or aperiodic transition appears, according to the surface treatment, the sample thickness, and the direction of the magnetic field. Hence the critical value of r can be controlled in a wide range.