The fact that the cross-profile of the glacial valley could be well approximated by parabolas ( Y = a x b , b = 2.0) is explained by the variation principle, assuming that the glacier erosion works towards minimizing the friction between ice and bedrock. The variation principle proves that the ideal or fully-developed morphology of the glacial valley should be a catenary, the curve which a chain hanging from two fixed points forms. Maclaurin's series expansion of the catenary equation shows that a parabola is a very good approximation of the catenary; hence, the good approximation of the cross-profile by parabolas.
Different catenaries are generated by changing the form ratio (depth/rim width) and are then approximated by Y = a X b by the method of least-squares. The b values obtained become only fractionally larger than 2.0 with increasing form ratios of up to 1.0, indicating that b values would range, in practice, between 1.0 and about 2.0. Two types of trend in the relationship between b values and the form ratio were obtained from several glaciers. For one type the b value becomes larger with increasing form ratios, and for the other the opposite. The first type is called the Rocky Mountain model after its source of data and represents overdeepening of the glacial valley development. The second type is called the Patagonia-Antarctica model, representing a widening, instead of a deepening, process of development. These differences are attributed to the nature of the glaciers which produced these valleys, i.e. alpine glaciers and continental ice sheets.