We describe the (alpha ,beta )-metrics whose the T-tensor vanishes (T-condition) and the (alpha ,beta )-metrics that satisfy the sigma T-condition sigma _hT^h_{ijk}=0, where sigma _h=frac{partial sigma }{partial x^h} and sigma is a smooth function on M. These classes have already been obtained by Shen and Asanov in a completely different approach. The Finsler metrics of the first class are Berwaldian, the metrics of the second class are almost regular non-Berwaldian Landsberg metrics.