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 of the question, we may be sure that $$i$$ varies directly as $$T$$ and inversely as $$r$$, and must therefore be proportional to $$T/\lambda^2r$$, $$T$$ being of three dimensions in space. In passing from one part of the spectrum to another $$\lambda$$ is the only quantity which varies, and we have the important law :—

When light is scattered by particles which are very small compared with any of the wave-lengths, the ratio of the amplitudes of the vibrations of the scattered and incident light varies inversely as the square of the wave-length, and the intensity of the lights themselves as the inverse fourth power.

I will now investigate the mathematical expression for the disturbance propagated in any direction from a small particle which a beam of light strikes.

Let the vibration corresponding to the incident light be expressed by $$A\cos(2\pi bt/\lambda)$$. The acceleration is

so that the force which would have to be applied to the parts where the density is $$D'$$, in order that the wave might pass on undisturbed, is, per unit of volume,

To obtain the total force which must be supposed to act over the space occupied by the particle, the factor $$T$$ must be introduced. The opposite of this conceived to act at $$O$$ (the position of the particle) gives the same disturbance in the medium as is actually caused by the presence of the particle. Suppose, now, that the ray is incident along $$OY$$, and that the direction of vibration makes an angle a with the axis of $$x$$, which is the line of the scattered ray under consideration—a supposition which involves no loss of generality, because of the symmetry which we have shown to exist round the line of action of the force. The question is now entirely reduced to the discovery of the disturbance produced in the aether by a given periodic force acting at a fixed point in it. In his valuable paper “On the Dynamical Theory of Diffraction”, Professor Stokes has given a complete investigation of this problem; and I might assume the result at once, The method there used is, however, for this particular purpose very indirect, and accordingly I have thought it advisable to give a comparatively short cut to the result, which will be found at the end of the present paper. It is proved that if the total force acting at $$O$$, in the manner supposed be $$F\cos(2\pi bt/\lambda)$$, the resulting disturbance in the ray propagated along $$OX$$ is