Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/273

If we write Curl for the differentiating vector operator which Maxwell calls by that name, equations (8) may be put in the form

$4\pi {\mathfrak {D}}={\text{Curl Curl }}(iu),$

whence

$d{\mathfrak {D}}/dt=(4\pi )^{-1}{\text{ Curl Curl }}(idu/dt).$

From $d{\mathfrak {D}}/dt$ we may calculate the magnetic induction ${\mathfrak {B}}$ by an operation which is the inverse of $(4\pi )^{-1}{\text{ Curl.}}$ We have therefore

${\mathfrak {B}}={\text{Curl }}(i\,du/dt),$

or

${\mathfrak {B}}=[r^{-3}{\text{F'}}(t-cr)+cr^{-2}{\text{F''}}(t-cr)](yk-zj).$

The magnetic induction is therefore zero except in the waves.

Equations (4) and (9) give the value of $d{\mathfrak {A}}/dt$ as function of $t$ and $r.$ By integration, we may find the value of ${\mathfrak {A}}$ Maxwell's "vector potential." This will be of the form of the second member of (4) multiplied by $-c^{-2}$ if we should give each ${\text{F}}$ one accent less, and for an unaccented ${\text{F}}$ should write ${\text{F}}_{1},$ to denote the primitive of ${\text{F}}$ which vanishes for the argument $\infty .$

That which seems most worthy of notice is that although simultaneously with the discharge of the system (A, B) the values of what we call the electric potential, the electrodynamic force of induction, and the "vector potential," are changed throughout all space, this does not appear connected with any physical change outside of the waves, which advance with the velocity of light.

If we now suppose that there is a second pair of charged spheres ($c,d$), as in the original problem, the discharge of this pair will evidently occur when the relaxation of electrical displacement reaches it. The time between the discharges is, therefore, by Maxwell's theory, the time required for light to pass from one pair to the other.

It may also be interesting to observe that in the axis of $x,$ on both sides of the origin, $x\rho =r^{2}i,$ and equation (4) reduces to

$4\pi {\mathfrak {D}}=[2r^{-3}{\text{F}}(t-cr)+2cr^{-2}{\text{F'}}(t-cr)]i.$

Here, therefore, the oscillations are normal to the wave-surfaces. This might seem to imply that plane waves of normal oscillations may be propagated, since we are accustomed to regard a part of an infinite sphere as equivalent to a part of an infinite plane. Of course, such a result would be contrary to Maxwell's theory. The paradox is explained if we consider that the parts of the wave-motion, expressed by ${\text{F}}$ and ${\text{F'}}$ diminish more rapidly than those expressed by ${\text{F''}},$ so that it is xmsafe to take the displacements in the axis of $x$ as approximately representing those at a moderate distance from it. In fact, if we consider the displacements not merely in the axis of $x,$ but within a cylinder about that axis, and follow the waves to an infinite distance from the origin, we find no approximation to what is usually meant by plane waves with normal oscillations.

J. Willard Gibbs.

New Haven, Conn., March 12 [1896].