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Owing to various departures from ideal simplicity, e.g. want of homogene in the primary vibrations, movement of the disturbing centres, the impo; bility of observing what takes place at a mathematical point, we are in eff only concerned with the average, and the average intensity is n times tl due to a single centre.
In the application to a cloud of acoustic resonators the restriction \ necessary that the resonators must not be close compared with X; otherw they would react upon one another too much. This restriction may app to exclude the case of the light from the sky, regarded as due mainly to 1 molecules of air; but these molecules are not resonators  at any rate regards visible radiations. We can most easily argue about an otherw uniform medium disturbed by numerous small obstacles composed oi medium of different quality. There is then no difficulty in supposing 1 obstacles so small that their mutual reaction may be neglected, even althon the average distance of immediate neighbours is much less than a wa length. When the obstacles are small enough, the whole energy dispers may be trifling, but it is well to observe that there must be some. '. medium can be -fully transparent in all directions to plane waves, wh is not itself quite uniform. Partial exceptions may occur, e.g. when the wi of uniformity is a stratification in plane strata. The dispersal then becon a regular reflexion, and this may vanish in certain cases, even though 1 changes of quality are sudden (black in Newton's rings)*. But such tra: parency is limited to certain directions of propagation,
To return to resonators : when they may be close together, we have consider their mutual reaction. For simplicity we will suppose that they lie on the same primary wave-front, so that as before in the neighbourly of each resonator we may take
Further, we suppose that all the resonators are similarly situated as rega: their neighbours, e.g., that they lie at the angular points of a regu polygon. The waves diverging from each have then the same expressi and altogether
f-ila\       -ikrz         ~\
where rlt rz, ... are the distances of the point where \Jr is measured from 1 various resonators, and a is a coefficient to be determined. The wh potential is < + ^, and it suffices to consider the state of things at the fi resonator. With sufficient approximation
See Proc. Roy. Soc. Vol. ixxxvi A, p. 207 (1912); [This volume, p. 77]. Weber, Leipzig (1912), p. 252; Jahresber. d. Deutschen Math. Ver. Bd. xxi. p. 241 (1918). The mathematics has a very wide scope.