Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/78

62 Now the supposed property of the direction of $$i$$ requires that when $$\rho$$ coincides with $$i$$ and $$d\rho$$ is perpendicular to $$i, d\rho '$$ shall be perpendicular to $$\rho ',$$ which will then be parallel to $$\alpha.$$ But if $$d\rho$$ is parallel to $$j$$ or $$k,$$ it will be perpendicular to $$i,$$ and $$d\rho '$$ will be parallel to $$\beta$$ or $$\gamma,$$ as the case may be. Therefore $$\beta$$ and $$\gamma$$ are perpendicular to $$\alpha.$$ In the same way it may be shown that the condition relative to $$j$$ requires that $$\gamma$$ shall be perpendicular to $$\beta.$$ We may therefore set where $$i', j', k',$$ like $$i, j, k,$$ constitute a normal system of unit vectors (see No. 11), and $$a, b, c$$ are scalars which may be either positive or negative.

It makes an important difference whether the number of these scalars which are negative is even or odd. If two are negative, say $$a$$ and $$b,$$ we may make them positive by reversing the directions of $$i'$$ and $$j'.$$ The vectors $$i', j', k'$$ will still constitute a normal system. But if we should reverse the directions of an odd number of these vectors, they would cease to constitute a normal system, and to be superposable upon the system $$i, j, k.$$ We may, however, always set either

with positive values of $$a, b,$$ and $$c.$$ At the limit between these cases are the planar dyadics, in which one of the three terms vanishes, and the dyadic reduces to the form in which $$a$$ and $$b$$ may always be made positive by giving the proper directions to $$i'$$ and $$j'.$$

If the numerical values of $$a, b, c$$ are all unequal, there will be only one way in which the value of $$\Phi$$ may be thus expressed. If they are not all unequal, there will be an infinite number of ways in which $$\Phi$$ may be thus expressed, in all of which the three scalar coefficients will have the same values with exception of the changes of signs mentioned above. If the three values are numerically identical, we may give to either system of normal vectors an arbitrary position.

136. It follows that any self-conjugate dyadic may be expressed in the form where $$i, j, k$$ are a normal system of unit vectors, and $$a, b, c$$ are positive or negative scalars.

137. Any dyadic may be divided into two parts, of which one shall be self-conjugate, and the other of the form $$\text{I} \times \alpha.$$ These parts are found by taking half the sum and half the difference of the dyadic and its conjugate. It is evident that