570 ON THE LIGHT EMITTED FROM A [4
I, m, n being the direction cosines of OQ. On the whole
— 4<TrR^ = 2, cos [pt -l- e - kR6 + Jc (Iso + my + m)}, ......(15)
in which RQ is a constant for all the sources, but e, x, y, z vary from one soui to another. The intensity in the direction (I, m, ri) is thus represented by
[2 cos {e + k (Ix + my + nz)}J + [2 sin {e + k (lx + my + nz)}]2, or by
n + 22 cos [ej — 62 + k [I (os-i — oc2) + m (yt — ya) + n (^ — zz)}], .. .(16)
the summation being for all the %n(n — I) pairs of sources. In order to fi the work done we have now to integrate (16) over angular space.
It will suffice if we effect the integration for the specimen term; and shall do this most easily if'we take the line through the points (a\, ylt i (#2,2/23 z%) as axis °f reference, the distance between them being denoted by If (I, m, n) make an angle with D whose cosine is p,
Dp = l((Ci — #2) + in (y-i — yz) + n (z^ — z^), ............(17)
and the value of the specimen term is
I cos (gj — e2 + kDp) dp, J -i that is
2 sin kD cos (ej — e2)
The mean value of (16) over angular space is thus
I ygsin&flcosfo-e,)
Id -TT ut-i 'i~~r\ i ..................... \ -1- o I
kD
where ej; e2 refer to any pair of sources and D denotes the distance betwe them. If all the sources are in the same initial phase, cos (^ — e2) = 1. If tl distance between every pair of sources is a multiple of £\, sin kD = 0, a] (19) reduces to its first term.
We fall back upon a former particular case if we suppose that there a only two sources and that they are in the same phase.
If the question of the phases of the two sources be left open, (19) gives
(20)
If D be small, this reduces to
2 + 2 cos (ej - es),
which is zero if the sources be in opposite phases, and is equal to 4 if i phases be the same. .
* In the paper referred to, equation (19), /j. was inadvertently used in two senses.points finitely distant from the origin 0. The velocity-potential of one of these at (so, y, z}, estimated at any point Q, is-