on the Undulatory Hypothesis of Light, 469 



arbitrary directions of the axes, and arbitrary values of m, X, and 

 c. A discussion of the function f s contained in art. 30 of the 

 communication in the Philosophical Magazine for August 

 1862, shows that there are positions of maximum condensation 

 and no transverse velocity, alternating with positions of no con- 

 densation and maximum transverse velocity, at fixed distances 

 from the axis, and that the maximum condensations and ve- 

 locities are continually less as the distances from the axis are 

 greater. From these preliminaries we may proceed to the con- 

 sideration of a case of motion bearing on the immediate object 

 of this inquiry. 



Let us conceive the fluid to be put in motion by a plane sur- 

 face of indefinite extent, vibrating perpendicularly to its plane 

 according to the law of the sine of a circular arc. It is evident 

 that the resulting motion of the fluid will be wholly in lines 

 perpendicular to the disturbing surface. But, from what has 

 just been argued, the motion is still composed of motions defined 

 by the foregoing equations. Hence, to satisfy the conditions of 

 the disturbance, these component motions must be unlimited 

 in number and iD the same phase of vibration, and their axes 

 must ail be perpendicular to the disturbing plane and be re- 

 gularly distributed relatively to that plane. For under these 

 circumstances there will be no reason why the transverse mo- 

 tion at a given point should be in one direction rather than 

 another, and consequently the transverse motions will be de- 

 stroyed. If we now conceive the fluid to be bounded by a plane 

 of unlimited extent, and a certain limited portion of it to vibrate 

 in the manner above specified, there will be transverse motions 

 to certain distances within and without the geometrical boundary 

 of the disturbance. But from the above-stated properties of the 

 function /, it may be seen that this motion will be restricted 

 within very narrow limits. For since the number of the axes of 

 the component motions within the geometrical boundary is unli- 

 mitedly great, at points very little distant from the boundary on 

 the outside the sum of the condensations cannot sensibly differ 

 from the sum of the rarefactions, and beyond that distance there 

 can be no sensible motion ; whilst at like distances within the. 

 boundary there will on the same account be no reason for a 

 transverse motion in one direction rather than in another, and 

 the motion will thus be wholly longitudinal. These considera- 

 tions enable us to perceive why the vibrations of an elastic fluid 

 do not spread indefinitely at their lateral boundaries, and why 

 a slender cylindrical portion of light may be propagated through 

 space to an unlimited distance. 



If the plane vibrated according to any other law, provided 

 there be no motion of translation, the motion may be regarded 



