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47. ^  _**a** _ _**b** _ Roberts, Fig. 17: "Effect of torque transmission on rolling resistance", p. 71
48. ** ^  ** Деев, p.30 including eq. (2.7) and Fig. 2.3
49. ** ^  ** Астахов, Figs. 3.8, 3.9, 3.11, pp. 50-55. Hay, Fig. 60-2, p. 72 shows the same phenomena but has higher values for Crr and not reported here since the railroads in 2011 [ [1] ](https://web.archive.org/web/20100626045718/http://www.istc.illinois.edu/about/SeminarPresentations/20091118.pdf) . were claiming about the same value as Астахов
50. ** ^  ** Hersey, Table 6., p. 267
51. ** ^  ** Per this assumption,  F  =  k  N  0.5  {\displaystyle F=kN^{0.5}}  ![{\\displaystyle F=kN^{0.5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7b6c9e14cf4741f040fc8fa66c3ff409250a1a) where  F  {\displaystyle F}  ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) is the rolling resistance force and  N  {\displaystyle N}  ![{\\displaystyle N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3) is the normal load force on the wheel due to vehicle weight, and  k  {\displaystyle k}  ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) is a constant. It can be readily shown by differentiation of  F  {\displaystyle F}  ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) with respect to  N  {\displaystyle N}  ![{\\displaystyle N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3) using [ this rule ](/wiki/Differentiation_rules