Mathematical Theory of Aerial Vibrations. 95 



of arbitrary disturbance. The explanation I give of those 

 contradictions is as follows. I regard them as proving, not 

 that cases of plane- waves and spherical waves cannot exist, 

 but that they cannot exist spontaneously. Having been led 

 deductively from the general equations to those which define 

 ray-vibrations, without meeting with any similar contradiction, 

 and anterior to any supposed case of motion, I conclude that 

 ray-vibrations are common to all instances of vibratory mo- 

 tion. Any instance of such motion may be conceived to be 

 composed of the ray-vibrations defined by the equations (2.) 

 and (3.), the number, magnitude, and directions of the rays 

 being unlimited. Accordingly if a plane-wave were generated, 

 it would be composed of an unlimited number of ray-vibrations 

 having their axes all parallel to the direction of propagation ; 

 and from what has been proved of these vibrations, the result- 

 ant wave might be propagated to any distance without under- 

 going any change. So if a spherical wave were generated, it 

 would be composed of an unlimited number of ray-vibrations 

 having their axes diverging from a centre, and the change of 

 condensation with the change of distance from the centre would 

 be according to the inverse square of the distance, since it 

 would depend only on the divergence of the rays. This re- 

 sult is in accordance with the principle of the constancy of 

 mass. 



Cambridge Observatory, 

 January 9, 1849. 



Postscript^ Jan. 11, 1849. — After despatching the foregoing 

 communication, I deduced from my theory of ray-vibrations 

 a result which ought, I think, to command the attention of 

 mathematicians. 



I have already exhibited the course of reasoning which 

 finally led to the conclusion, that the transverse motion in ray- 

 vibrations is defined, to the first order of approximation, by 

 the foregoing equation (2.). lo this equation is next to be 

 applied the process by which, from the equation (3.) of ana- 

 logous form, which defines the motion along the axis of the 

 ray, a unique integral expressed in finite terms was obtained, 

 viz. 



27r/ , /\ e\^\ 



f = 7n cos — (s-a/w H — j-l. 



(See Phil. Mag. for April 1818, p. 279.) On so doing, the 

 result is 



/=» cos {gx + hy), 



