Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/42

We may infer from the similarity of these equations that the relations of $\alpha ,\beta ,\gamma$, and $\alpha ',\beta ',\gamma '$ are reciprocal, a proposition which is easily proved directly. For the equations

$\alpha ={\frac {\beta '\times \gamma '}{\alpha '.\beta '\times \gamma '}},$⁠$\beta ={\frac {\gamma '\times \alpha '}{\beta '.\gamma '\times \alpha '}},$⁠$\gamma ={\frac {\alpha '\times \beta '}{\gamma '.\alpha '\times \beta '}}$

(4)

are satisfied identically by the substitution of the values of $\alpha ',\beta '$, and $\gamma '$ given in equations (1). (See Nos. 31 and 34.)

Def.—It will be convenient to use the term reciprocal to designate these relations, i.e., we shall say that three vectors are reciprocals of three others, when they satisfy relations similar to those expressed in equations (1) or (4).

With this underatanding we may say:—

The coefficients by which any vector is expressed in terms of three other vectors are the direct products of that vector with the reciprocals of the three.

Among other relations which are satisfied by reciprocal systems of vectors are the following:

$\alpha .\alpha '=\beta .\beta '=\gamma .\gamma '=1,$

$\alpha .\beta =0,$⁠$\alpha .\gamma '=0,$⁠$\beta .\alpha '=0,$⁠$\beta .\gamma '=0,$⁠$\gamma .\alpha '=0,$⁠$\gamma .\beta '=0.$

(5)

These nine equations may be regarded as defining the relations between $\alpha ,\beta ,\gamma$, and $\alpha ',\beta ',\gamma '$ as reciprocals.

$(\alpha .\beta \times \gamma )(\alpha '.\beta '\times \gamma ')=1.$

(6)

(See No. 34.)

$\alpha \times \alpha '+\beta \times \beta '+\gamma \times \gamma '=0.$

(7)

(See No. 29.)

A system of three mutually perpendicular unit vectors is reciprocal to itself, and only such a system. The identical equation

$\rho =(\rho .i)i+(\rho .j)j+(\rho .k)k$

(8)

may be regarded as a particular case of equation (2).

The system reciprocal to $\alpha \times \beta ,\beta \times \gamma ,\gamma \times \alpha$ is

$\alpha '\times \beta ',$⁠$\beta '\times \gamma ',$⁠$\gamma '\times \alpha ',$

or

${\frac {\alpha }{\alpha .\beta \times \gamma }},$⁠${\frac {\beta }{\alpha .\beta \times \gamma }},$⁠${\frac {\gamma }{\alpha .\beta \times \gamma }}\cdot$

38a. If we multiply the identical equation (8) of No. 37 by $\sigma \times \tau$, we obtain the equation

$(\alpha .\beta \times \gamma )(\rho .\sigma \times \tau )=\alpha .\rho (\beta .\sigma \,\gamma .\tau -\beta .\tau \,\gamma .\sigma )+\beta .\rho (\gamma .\sigma \,\alpha .\tau -\gamma .\tau \,\alpha .\sigma )+\gamma .\rho (\alpha .\sigma \,\beta .\tau -\alpha .\tau \,\beta .\sigma ),$

which is therefore identical. But this equation cannot subsist identically, unless

$(\alpha .\beta \times \gamma )\sigma \times \tau =\alpha (\beta .\sigma \,\gamma .\tau -\beta .\tau \,\gamma .\sigma )+\beta (\gamma .\sigma \,\alpha .\tau -\gamma .\tau \,\alpha .\sigma )+\gamma (\alpha .\sigma \,\beta .\tau -\alpha .\tau \,\beta .\sigma )$