Variational calculus methods are applied to the problem of dispersion of mantle Rayleigh waves. In the present paper we have worked two models. One is Gutenberg's model with a low-velocity layer around 150 km. depth. The other is a Jeffreys-Bullen model modified above 200 km. depth so as to join smoothly to the explosion-determined velocities just under the Mohorovičić discontinuity. No low-velocity layer is assumed in this model. Both models give almost identical theoretical dispersion curves which agree well with the Ewing-Press observations of mantle Rayleigh waves for periods longer than 250 sec. This result means that the minimum group velocity at about 250 sec. is mainly due to a sharp increase of shear velocity at about 400 km. depth, which is a common feature for the two models. For periods shorter than 250 sec. Gutenberg's model gives results concordant with the observations. The modified Jeffreys-Bullen model disagrees significantly with the observations. This demonstrates the existence of a low-velocity layer in the upper mantle.