Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/235

the real part represents the averaged displacement $[{\mathsf {U}}]_{\text{Ave}},$ and the coefficient of $\iota$ the rate of increase of the same multiplied by a constant factor. This bivector therefore represents the average state of a small part of the field both with respect to position and velocity. We may also say that the coefficient of $\iota$ in ${\mathsf {W}}$ represents the value of the averaged displacement $[{\mathsf {U}}]_{\text{Ave}}$ at a time one-quarter of a vibration earlier than the time principally considered.

11. It may serve to fix our ideas to see how ${\mathsf {W}}$ is expressed as a function of the time. We may evidently set

$[{\mathsf {U}}]_{\text{Ave}}={\mathsf {A}}_{1}\cos {\frac {2\pi }{p}}t+{\mathsf {A}}_{2}\sin {\frac {2\pi }{p}}t$

where ${\mathsf {A}}_{1}$ and ${\mathsf {A}}_{2}$ are vectors representing the amplitudes of the two parts into which the vibration is resolved. Then

${\frac {p}{2\pi }}[{\dot {\mathsf {U}}}]_{\text{Ave}}=-{\mathsf {A}}_{1}\sin {\frac {2\pi }{p}}t+{\mathsf {A}}_{2}\cos {\frac {2\pi }{p}}t,$

and

$[{\mathsf {U}}]_{\text{Ave}}-\iota {\frac {p}{2\pi }}[{\dot {\mathsf {U}}}]_{\text{Ave}}=({\mathsf {A}}_{1}-\iota {\mathsf {A}}_{2})\left(\cos {\frac {2\pi }{p}}t+\iota \sin {\frac {2\pi }{p}}t\right);$

that is, if we set ${\mathsf {A}}={\mathsf {A}}_{1}-\iota {\mathsf {A}}_{2},$

${\mathsf {W}}={\mathsf {A}}e^{\frac {2\pi \iota t}{p}}.$

(13)

In like manner we may obtain

${\text{Q}}=ge^{\frac {2\pi \iota t}{p}},$

(14)

where $g$ is a biscalar, or complex quantity of ordinary algebra. Substituting these values in (12), and cancelling the common factor containing the time, we have

${\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}{\mathsf {A}}-\nabla g=\Theta {\mathsf {A}}.$

(15)

Our equation is thus reduced to one between ${\mathsf {A}}$ and $g,$ and may easily be reduced to one in ${\mathsf {A}}$ alone.^[1] Now ${\mathsf {A}}$ represents six numerical quantities (viz., the three components of ${\mathsf {A}}_{1},$ and the three of ${\mathsf {A}}_{2}$), which may be called the six components of amplitude. The equation, therefore, substantially represents the relations between the six components of amplitude in different parts of the field.^[2] The equation is, however, not really different from (12), since ${\mathsf {A}}$ and $g$ are only particular values of ${\mathsf {W}}$ and ${\text{Q}}.$

1. The terms $\nabla {\text{Q}},\nabla q$ are allowed to remain in these equations, because the best manner of eliminating them will depend somewhat upon our admission or rejeotion of the solenoidal hypothesis.
2. The representation of the six components of amplitude by a single letter should not be regarded as an analytical artifice. It only leaves undivided in our notation that which is undivided in the nature of things. The separation of the six components of amplitude is artificial, in that it introduces arbitrary elements into the discussion, viz. the directions of the axes of the coordinates, and the zero of time.