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1918]                 RANDOM DISTRIBUTION OF  LUMINOUS  SOURCES                    569
When there are two unit sources distant D from one another and in the same initial phase, the potentials may be taken to be
.         cos k (at  r)         ,         cos k (at - r)                 ._ - N
9 =-----------p----------,      w=-----------:;-------..........(11)
T-                 /i,                 r                 4>7rr
At the first source where r = 0
4nrr-d(j)/dr = cos kat  kr sin kat,
dd>    d^r    ka sin kat      ka    .   7 , .     .
-rr H77 = T----------1- , n sin A; (ac  1A
cw       cw            477T         4f7rD
The work done by the source at r - 0 is accordingly proportional to
l + S~,    ..............................(12)
and an equal amount of work is done by the source at ?'' = 0. If D be infinitely great, the sources act independently, and thus the scale of measurement in (12) is such that unity represents the work done by each source when isolated. If D = 0, the work done by each source is doubled, and the sources become equivalent to one of doubled magnitude.
If D be equal to ^X, or to any multiple thereof, sinfcD == 0, and we see from (12) that the work done by each source is unaffected by the presence of the other. This conclusion may be generalized. If any number (?i) of equal sources in the same phase be arranged in (say a vertical) line so that the distance between immediate neighbours is X, the work done by each is the same as if the others did not exist. The whole work accordingly is n, whereas the work to be done by a single source of magnitude n would be n2. Thus if sound be wanted only in the horizontal plane where there is agreement of phase, the distribution into n parts effects an economy in the proportion of nil.
A similar calculation would apply when the initial phases differ, but we will now take up the problem in a more general form where there are any number (n) of unit sources, and by another method*. The various centres are situated at points finitely distant from the origin 0. The velocity-potential of one of these at (so, y, z}, estimated at any point Q, is-
fk   cos (P*+ e ~~ ^)                                    (V\\
where R is the distance between Q and (x, y, z). At a great distance from the origin we may identify R in the denominator with OQ, or R0; while under the cosine we write
R = R0  (loo + my + nz\ ........................(14)
* " On the Production and Distribution of Sound," Phil. Mag. Vol. vi. p. 289 (1903); Scientific Papers, Vol. v. p. 136.are aiming