Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/85

Rh Def.—The displacement represented by the equation is called inversion. The most general case of a homogeneous strain may therefore be produced by a pure strain and a rotation with or without inversion.

150. If

The general problem of the determination of the principal ratios and axes of strain for a given dyadic may thus be reduced to the case of a right tensor.

151. Def.—The effect of a prefactor of the form where $$a, b, c$$ are positive or negative scalars, $$\alpha, \beta, \gamma$$ non-complanar vectors, and $$\alpha ', \beta ', \gamma '$$ their reciprocals, is to change $$\alpha$$ into $$a\alpha, \beta$$ into $$b\beta,$$ and $$\gamma$$ into $$c\gamma.$$ As a postfactor, the same dyadic will change $$\alpha '$$ into $$a\alpha ', \beta '$$ into $$b\beta ',$$ and $$\gamma$$ into $$c\gamma '.$$ Dyadics which can be reduced to this form we shall call tonic (Gr. ). The right tensor already described constitutes a particular case, distinguished by perpendicular axes and positive values of the coefficients $$a, b, c.$$

The value of the dyadic is evidently not affected by substituting vectors of different lengths but the same or opposite directions for $$\alpha, \beta, \gamma,$$ with the necessaiy changes in the values of $$\alpha ', \beta ', \gamma ',$$ defined as reciprocals of $$\alpha, \beta, \gamma.$$ But, except this change, if $$a, b, c$$ are unequal, the dyadic can be expressed only in one way in the above form. If, however, two of these coefficients are equal, say $$\alpha$$ and $$\beta,$$ any two non-collinear vectors in the $$\alpha\text{-}\beta$$ plane may be substituted for $$\alpha$$ and $$\beta,$$ or, if the three coefficients are equal, any three non-complanar vectors may be substituted for $$\alpha, \beta, \gamma.$$

152. Tonics having the same axes (determined by the directions of $$\alpha, \beta, \gamma$$) are homologous, and their multiplication is effected by multiplying their coefficients. Thus, Hence, division of such dyadics is effected by division of their coefficients. A tonic of which the three coefficients $$a, b, c$$ are unequal, is homologous only with such dyadics as can be obtained by varying the coefficients.

153. The effect of a prefactor of the form or