This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 edition. Excerpt: ...research for the expression of the velocities of the component system of relative motion, we have for the mean inequalities referred to their principal axes, where 2a//qr is the time-mean of the velocities of a grain, and X is the measure of the scale of the system of relative motion. (N.B. These rates ...
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 edition. Excerpt: ...research for the expression of the velocities of the component system of relative motion, we have for the mean inequalities referred to their principal axes, where 2a//qr is the time-mean of the velocities of a grain, and X is the measure of the scale of the system of relative motion. (N.B. These rates are independent of a-.) As already pointed out, Art. 146, the expressions in equations (189) and (190) for the inequalities are with reference to their principal axes only; so that in order to obtain expressions that shall apply for any axes it is necessary to effect the transformation from the principal axes, at a point, to fixed axes. 152. Rates of angular dispersion referred to axes which are not necessarily principal axes of rates of distortion. Taking l, m, n, l, m, n, l, m, n, to be respectively the direction cosines of the principal axes with reference to any rectangular system of fixed axes, at, bl, 0/, fl, 9/, hi to be the mean values of u," 0," w," 1/w', w'u', u'v' (u', &c., as before, representing the relative velocities referred to the principal axes 1, 2, 3), and let Also since the principal axes do not change their position in consequence of the dispersion of the inequalities Then substituting from equations (190) for 8,11'/8., t, &c., in (194), and remembering that Z11''+ l, v' + law', when referred to the principal axes is the same as u' referred to the fixed axes, we have by equation (193), for the rates of dispersion, referred to any axes, 153. The analytical definition of the rates of angular redistribution qf inequalities in rates of conduction through the grains. As already proved, Arts. 78 C and 79, Section VII., and the theorem Art. 136 in this section, ...
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