576 ON THE LIGHT EMITTED FROM A [436 We thus obtain for (34) [cos kR' {(h* + 1) kR' + 2/i] + smkR {(A» + l) hkR + /ia-l}] ~r + _ 47r, , [cos fcJR {(/i,2 + 1) IcR + 2A} + sin fcJB {(^ + 1) hkR + h*- I}]. /c3(i + /IT ...... (35) When we combine the first and third parts, in which R' does not appear, we get + sin &.Z2 {/i4 + 3A2 + A (A2 + 1) kR}]. _ Is ( 1 4" /i )~ ...... (36) The first part of (35), representing the effect due to the sphere R suddenly terminated, is of order kR ; and our object is to ascertain whether by suitable' choice of h and R' we can secure the relative annulment of (35). As regards (36), it suffices to suppose h small enough. In the- second part of (35) the principal term is of relative order (R'/R^e-M'W-W anc[ can be annulled by sufficiently increasing R', however small h may be. Suppose, to take a numerical example, that h jifaT5> and tha^ e~Jl/ct-R'~2i'> is also -. Then - 27rA Iog10 e h With such a value of R' R the factor R'/R may be disregarded *. It appears then that it is quite legitimate to regard the intensity due to the simple sphere, expressed in (33), as a surface effect ; and this conclusion may be extended to the corresponding term involving n2 in (28), relating to discrete centres scattered at. random. This extension being important, it may be well to illustrate it further. Returning to the consideration of n sources in the same initial phase distributed at random along a limited straight line, let us inquire what is to be expected at a distant point along the line produced. The first question which suggests itself is Are the phases on arrival distributed at random ? Not in all cases, but only when the limited line contains exactly an integral number of wavelengths. Then the phases on arrival are absolutely at random over the whole period, and accordingly the expectation of intensity is n precisely. If, however, there be a fractional part of a wave-length outstanding, the arrival phases are no longer absolutely at random, and the conclusion that the expectation of intensity is n simply cannot be maintained. Suppose further that n is so great that the average distance between consecutive sources is a very small * The application to light is here especially in view.member reduces to n2.