another point of ds, as the vertex of the pencil, we arrive at a somewhat different element on the surface s Thus the rays, which ds, receives from s, through different points on ds, do not all issue from one and the same element

of Se

Since however the area of ds, may be any whatever, nothing hinders us from supposing it so small, that it is an indefinitely small quantity of a higher order than dsɑ. In this case, if the vertex of the pencil changes its position within ds,, then the element of s, which corresponds to ds will change its position through a distance so small that in comparison with the dimensions of the element it is indefinitely small and may be neglected. Hence in this case the element ds, which we obtain when we choose any point whatever p, on element ds,, and make it the vertex of the pencil of rays issuing from ds, may be considered as the part of ds, which exchanges rays with ds, through ds. The area of this element ds, is easily found from what precedes. Let us suppose as before that a tangent plane to the surface s is drawn at p, and that tangent planes to the surfaces s and s are drawn at points on the elements ds and ds, respectively; and let us consider the two latter elements as elements of the tangent planes. Take systems of co-ordinates on these three tangent planes, and form the quantities A and C, as given by the first and third of equations (I.). Then by equation (II.) we have

a

The quantity of heat which ds, sends to ds, and which, as mentioned above, may be considered as the quantity which ds, receives from the surface s through ds,, is expressed by

a

or, substituting for ds, its value given above,

2

If we compare this expression with that found above, which expresses the quantity of heat sent by ds, through ds, to s., we see that the two stand to each other in the ratio ev: ev. If we now suppose that s and s, are the surfaces of two perfectly black bodies of equal temperature, and make for such surfaces the assumption (which we have already seen to be necessary in the case of radiation without concentration), that the two products ev and ev are equal, then the quantities of heat given by the two expressions above are also equal.

2

§ 17. Mutual Radiation of Entire Surfaces.

If on the intermediate surface s, we take, instead of the element hitherto considered, another element which is also supposed to be an indefinitely small quantity of a higher order, then the element of s., which exchanges rays with ds. through this element of s,, has a different position from that in the last case; but the two quantities of heat thus exchanged are again equal to each other; and the same holds of all other elements of the intermediate surface.

To obtain the quantity of heat which ds, sends to s, not through a single element of the intermediate surface, but as a whole, and similarly the quantity of heat which as a whole it receives from s, we must integrate the two expressions found above over the surface s,, so far as this surface is cut by the rays which pass from ds to s, and vice versâ. It is evident that if for each element of surface ds, the two differentials are equal, then the whole integrals must also be equal.

Lastly, to find the quantities of heat, which the whole surfaces exchanges with s., we must integrate both these expressions over the surface s. This process again will not disturb the equality, which exists for each of the separate elements ds..

Thus the principle previously discovered in a special case, viz. that two perfectly black bodies of equal temperatures exchange equal quantities of heat with each other, so far as the equation evev holds for them, appears also as the result of an investigation, which in no way depends upon whether the rays issuing from s suffer a concentration at se, and vice versâ, or not; since the only condition was, that the rays

issuing from 8 and s, suffer no concentration at the intermediate surface 8, a condition which may be always fulfilled, since this surface may be chosen at pleasure.

It follows further from this result that, if a given black body exchanges heat not only with one but with any number of black bodies, it receives from all of them exactly the same quantity of heat as it sends to them.

§ 18. Consideration of Various Collateral Circumstances.

The previous investigation has been made throughout under the assumption that any reflections and refractions take place without loss, or that there is no absorption of heat. We can, however, easily go on to shew that the results are not altered, if this condition is dropped.. For consider any one of the different processes, by which a ray may be weakened on its way from one body to another; say that at a place where the ray cuts the boundary of two media, one part passes into the further medium by refraction, and the other is reflected. Then whether we consider the one part or the other as the continuation of the original ray, we have in both cases a weakened ray to deal with. The same holds if a ray be partially absorbed by entering a medium. But in each of these cases we have the law that two rays which traverse the same path in opposite directions are weakened in equal proportion. The quantities of heat, which two bodies mutually send to each other, are therefore weakened by such processes to the same extent; so that, if they would be equal without such weakening, they will also be equal when thus weakened. Another circumstance may be considered in connection with the processes above mentioned, viz. that a body may receive from the same direction rays which proceed from different bodies. For example, a body A may receive from a point, which lies on the bounding surface of two media, two rays coinciding in direction, but issuing from two different bodies B and C. One of these may come from the bounding medium and be refracted at the point, whilst the other is already in the bounded medium, and is reflected at the point. In this case, however, the two rays are weakened by refraction and reflection in such a way, that, if both were before of equal intensity, their sum afterwards is

of the same intensity as either one of them had beforehand. Now suppose a ray of the same intensity to issue from the body A in the opposite direction, this will be divided, at the same point, into two parts, of which one enters the bounding medium, and passes forward to the body B, while the other is reflected and passes to the body C. The two parts which thus reach B and C from A are exactly as great as those which A receives from B and C. The body A thus stands to each of the bodies B and C in such a relation, that, assuming equal temperatures, it exchanges with them equal quantities of heat. The equality of the modifications which two rays undergo, when passing in opposite directions in any path whatever, must produce the same result in all other cases however complicated.

Again if, instead of perfectly black bodies, we consider such as only partially absorb the rays falling on them; or if instead of homogeneous heat we consider heat which contains systems of waves of different lengths; or lastly, if instead of taking all the rays as unpolarized we include the phenomena of polarization; still in all these cases we have to do only with facts, which hold equally for the heat sent out by any one body, and for that which it receives from other bodies. It is not necessary to consider these facts more closely, since they also take place with ordinary radiation without concentration, and the object of the present investigation was only to consider the special actions which might possibly be produced by concentration of the rays.

§ 19. Summary of Results.

The main results of this investigation may be briefly

stated as follows:

(1) In order to bring the action of ordinary radiation, without concentration, into accordance with the fundamental principle, that heat cannot of itself pass from a colder to a hotter body, it is necessary to assume that the intensity of emission from a body depends not only on its own composition and temperature, but also on the nature of the surrounding medium; the relation being such, that the intensities of emission in different media stand in the inverse ratio of the squares of the velocities of radiation in the

media, or in the direct ratio of the squares of the coefficients of refraction.

(2) If this assumption as to the influence of the surrounding media is correct, the above fundamental principle is not only fulfilled in the case of radiation without concentration, but must also hold good when the rays are concentrated in any way whatever by reflections or refractions; since this concentration may indeed change the absolute magnitudes of the quantities of heat, which two bodies radiate to each other, but not the ratio between these quantities.