This proof relates only to the total radiation, and the index n was assumed constant for all wave lengths. But equation (77) holds also for the partial radiations of any one particular period T. Let the intensity of emission of P for rays whose periods lie between 7" and T -j- ^rbe denoted by i^dT. Similarly denote the intensity of radiation from P' for the same rays by i'TdT. Then, from (16),
*"/
/ 2
sin 0 cos 0(i ~ **X0 = o. . (i 8)
The 2 is to be extended over all periods between T = o and T= oo.
Between the two bodies P and P' conceive a layer introduced which is transparent to a certain wave length A, but reflects other wave lengths. Equation (18) must always hold, but the functional relation between r$ and T varies according to the thickness ancl nature of the layer. Now in order that (i 8) may hold as r$ is indefinitely varied, every term of the 2 in (18) must vanish, i.e. for every value of T*
z'T : iT = n* ....... (19)
According to Kirchhoff's law (9'), for a body which is not black the ratio of the emission z\ to the absorption a^ is proportional to the square of the index n of the surrounding medium. Since the change of aK with n may be calculated from the reflection equations, the relation between iK and n is at once obtained. In any case, then, for bodies that are not black the intensity of radiation is not strictly proportional to n*.
7. The Sine Law in the Formation of Optical Images of Surface Elements. — If ds' is the optical image of a surface element ds formed by a bundle of rays which are symmetrical
* Equation (17) can also be obtained by the method employed on page 497 if the space outside of the hollow sphere be conceived as filled with a medium different from that inside the sphere, but the calculation is somewhat more complicated. Since in such an. arrangement the waves of different periods T may be separated from one another by refraction and diffraction, (19) results at once from (17) in consideration of the conclusions upon page 497. the