Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/84

${\text{I}}$, or at least capable of expression with any required degree of accuracy as a root of ${\text{I}}.$

It should be observed that the value of the above dyadic will not be altered by the substitution for $\alpha$ of any other parallel vector, or for $\beta$ and $\gamma$ of any other conjugate semi-diameters (which succeed one another in the same angular direction) of the same or any similar and similarly situated ellipse, with the changes which these substitutions require in the values of $\alpha ',\beta ',\gamma '.$ Or, to consider the same changes from another point of view, the value of the dyadic will not be altered by the substitution for $\alpha '$ of any other parallel vector or for $\beta '$ and $\gamma '$ of any other conjugate semi-diameters (which succeed one another in the same angular direction) of the same or any similar and similarly situated ellipse, with the changes which these substitutions require in the values of $\alpha ,\beta ,$ and $\gamma ,$ defined as reciprocals of $\alpha ',\beta ',\gamma '.$

148. The strain represented by the equation

$\rho '=\{aii+bjj+ckk\}.\rho$

where $a,b,c$ are positive scalars, may be described as consisting of three elongations (or contractions) parallel to the axes $i,j,k,$ which are called the principal axes of the strain, and which have the property that their directions are not affected by the strain. The scalars $a,b,c$ are called the principal ratios of elongation. (When one of these is less than unity, it represents a contraction.) The order of the three elongations is immaterial, since the original dyadic is equal to the product of the three dyadics

$aii+jj+kk,$$ii+bjj+kk,$$ii+jj+ckk,$

taken in any order.

Def.—A dyadic which is reducible to this form we shall call a right tensor. The displacement represented by a right tensor is called a pure strain. A right tensor is evidently self-conjugate.

149. We have seen (No. 135) that eveiy dyadic may be expressed in the form

$\pm \{ai'i+bj'j+ck'k\},$

where $a,b,c$ are positive scalars. This is equivalent to

$\pm \{ai'i'+bj'j'+ck'k'\}$

and to

$\pm \{i'i+j'j+k'k\}.\{aii+bjj+ckk\}.$

Hence every dyadic may be expressed as the product of a versor and a right tensor with the scalar factor ± 1. The versor may precede or follow. It will be the same versor in either case, and the ratios of elongation will be the same; but the position of the principal axes of the tensor will difier in the two cases, either system being derived from the other by multiplication by the versor.