﻿628 Dr. C. V. Burton on the Scattering and 



be described ; the plane w= —/touches this sphere, and over 

 the small area effectively covered by the diffraction pattern 

 the sphere and the plane are indistinguishable. Excluding 

 from consideration that small area, and a like area at the 

 opposite pole (/, 0, 0), take some definite point P on the 

 surface of the sphere. If, then, Q is any point within 

 the square b X b, the length PQ may lie anywhere between 

 limits which differ by a large number of wave-lengths ; and 

 if a vibrator placed at Q is sending forth a disturbance (4) of 

 prescribed phase, the phase in which this disturbance reaches 

 P may be any whatever : under the conditions of the problem 

 all phases are equally likely, and this is true for each vibrator 

 independently of the other vibrators. Hence it follows that, 

 on an average, the (amplitude) 2 at the point P is equal to a 2 

 (that due to a single vibrator) multiplied by crlP the number 

 of vibrators *. The expression ah* 1 a 2 , being constant over 

 practically the whole spherical surface, has only to be 

 multiplied by 47r/ 2 to furnish the total diffusely radiated 

 energy, on the arbitrary scale already used. Thus, finally, 

 the plane-wave energy, reckoned in one direction only, bears 

 to the scattered energy the ratio 



<r 262 a 2X 2 / 2 /47rer6 2 a 2 / 2 =«r\ 2 /47r=7r<r/^. ... (5) 



8. No matter how thinly the plane of yz is besprinkled 

 with vibrators, this result is perfectly definite, provided only 

 that we can deal with a sufficiently extended area to be able 

 to assign a definite value to a. We are led to the conclusion 

 that the motion given out by the vibrators can be sharply 

 divided into two categories : plane waves and irregular 

 disturbance. If the vibrators are restricted to a finite area, 

 the plane waves and the irregular disturbance become sorted 

 out from one another at great distances, or at a more 

 moderate distance with the help of a lens. Their energies 

 are, of course, simply additive. 



9. It is otherwise evident that a portion of the resultant 

 disturbance from a plane distribution of synchronous vibrators 

 must be assignable to plane waves of unique specification. 

 For if the activity of the vibrators is due to the incidence 

 of primary plane waves, these waves must pass on with 

 diminished amplitude, and (in general) with altered phase ; 

 from a knowledge of which things the amplitude and phase 

 of the secondary plane waves emitted by the vibrators could 

 be written down. From considerations of symmetry we 

 should then know likewise the amplitude and phase of the 



* Of. Rayleigli, " Wave Tkeorv of Lip^t," § 4. Encycl. Brit. vol. xxiv. 

 (1888J ; Collected Papers, vol. iii". art, 148. 



