776 
MR. J. LARMOR OA" A DYNAMICAL THEORY OF 
movement of tlie slab, though tliere is no loss of energy ; thus, on direct refraction 
into a slab moving away from the light with velocity v, 
11 
fX -)■ i 
/X + 1 
3 _L\ 
•2 y J- 
If therefore Fresnel’s law is not fultilled, it would apparently be possible to con¬ 
centrate the radiation from the walls of an enclosure of uniform temperature by a self¬ 
acting arrangement of moving screens and transparent bodies inside the enclosure ; 
and this would be in contradiction to the Second Law of Thermodyamics.^'' 
78. The whole theory of rays is derived from the existence of the Hamiltonian 
characteristic function U, the })ath of a ray from one point to another in an isotropic 
medium being the course which makes SU or Sj'gc/s null, where /r is a function of 
position which is equal to the reciprocal of the efiective velocity of the light. The 
general law of illumination may be shown to follow from this, that if two elements of 
surface A and B are radiating to each other across any transparent media, the amount 
.of the radiation from A that is received by B is equal to the amount of radiation from 
B that is received by A; with the proviso, when different media are just in front of 
.1 and B, that the radiation of a body is cceteris iKtrihus to be taken as proportional 
to the square of the refractive index of the medium into which it radiates. Xow if 
that part v of the velocity of the light, which is pioduced by motion Through the 
medium of the bodies contained in it, make an angle 9 with the element of path ch, 
this equation will assume, after H. A. Lorentz and O. J. Lodge,^ the form 
S J (V + V cos 9)~^ ds = 0, 
which is to a first approximation 
S j ds -f- 81 V“^ (u dx -h V dy -f- lv dz) = 0, 
where V is the ordinary velocity of the light, and [u, v, w) are the components of u. 
In order that the paths of the rays in a homogeneous isotropic moving medium may 
remain the same as when the medium is at rest, the additional terms in the charac¬ 
teristic function must depend only on the limits of the integral, and therefore 
i( dx + V dy -h IV dz must be an exact differential; that is, the part thus added to 
the velocity of the light must be of irrotational character. If this part of the velocity 
were rotational, the law of illumination would not hold, as the type of the charac¬ 
teristic equation of the rays would thereby be changed. Tlius the equilibrium of 
exchanges of radiation which would subsist in an enclosure with the free mther in it 
* Cf. Clausius, “ On the Concentration of Rays of Light and Heat, and on the Limits of its A'.'tioii,” 
■ Papers on the Mechanical Theory of Heat,’ translated by W. R. Browxu, p[). 2'd5-33l. 
t 0. J. Lodgl, “ Aberi'ation Problems,” ‘Phil. Trans.,’ A, 1833, pp. 748-753. 
