1915] RESONATORS EXPOSED TO PRIMARY PLANE WAVES 281 If we write then A=-k-*ama...............................(9) So far a is arbitrary, since we have used no other condition than that no work is being done at the resonator. For instance, (9) applies when the source of disturbance is merely the presence at the origin of a small quantity of gas of varied character. The peculiar action of a resonator is to make A a maximum, so that sin a = + 1, say — 1. Then A^l/k, a = -i/k, ........................(10) • i"Ki« and As in (3), 47rr2Mod2^ = 47r//c2 = X2/vr, ..................... (12)- and the whole energy dispersed corresponds to an area of primary wave-front equal to X2/?r. The condition of resonance implies a definite relation between ((/> + ty) and d ($ + ty) / dr. If we introduce the value of a from (10), we see that this is _ •&) I dr~ " -l/ra ~ and this is the relation which must hold at a resonator so tuned as to respond to the primary waves, when isolated from all other influences. The above calculation relates to the case of a single resonator. For many purposes, especially in Optics, it would be desirable to understand the operation of a company of resonators. A strict investigation of this question requires us to consider each resonator as under the influence, not only of the primary waves, but also of the secondary waves dispersed by its neighbours, and in this many difficulties are encountered. If, however, the resonators are not too near one another, or too numerous, they may be supposed to act independently. From (11) it will be seen that the standard of distance is the wave-length. The action of a number (n) of similar and irregularly situated centres of secondary disturbance has been considered in various papers on the light from the sky*. The phase of the disturbance from a single centre, as it reaches a distant point, depends of course upon this distance and upon the situation of the centre along the primary rays. If all the circumstances are accurately prescribed, we can calculate the aggregate effect at a distant point, and the resultant intensity may be anything between 0 and that corresponding to complete agreement of phase among all the components. But such a calculation would have little significance for our present purpose. * Compare also "Wave Theory of Light," Enc. Brit. Vol. xxiv. (1888), § 4; Scientific Papers, Vol. m. pp. 53, 54.ift H. Weber, Leipzig (1912), p. 252; Jahresber. d. Deutschen Math. Ver. Bd. xxi. p. 241 (1918). The mathematics has a very wide scope.