UNDULATION —PRINCIPLE OF IIUYGIIENS. 157 



body continue, undulations will continue to flow ofiffrom'it tlirougli the air; but 

 that ihc moment they cease, the uiidiihitions will cease also. Also that during 

 the existence of any undulation there are actually and simultaneously existing^ 

 somewhere in its length, all the various velocities, positive and negative, which 

 belong successively, and one at a time, to the vihrating body. Also that, as tbc 

 ])rogress of the undulation is uniform, its fractional parts are proportional in 

 length to the times of performing ihem; and finally that the molecular velocity 

 at any point may be represented by observing to what fractional part of a avIioIc 

 undulation that point corresponds, and taking the sine of the same fraction of 

 an entire circle multiplied by the maximum velocity. 



It must not be understood, because, in what has been said, mention has only 

 been made of tremors in the direction of the original vibration, that therefore 

 undulations are confmed to that direction. It is impossible to disturb the equi- 

 librium of the molecules of an elastic fluid, at any point or in any direction 

 whatcv(!r, without disturbing that of all the adjacent molecules, and giving rise 

 to tr(!mors in all directions. At a distance from the vibrating body which is 

 considerable compared with the size of tin; body, therefore, the front of the 

 wave will be sensibly spherical, and when the body is small it may be regarded 

 as strict-ly so. 



It is furthermore manifest that the molecular movements need not be, as wo 

 have hitherto supposed them, linear, nor directed to and from the ct'utre of the 

 wave. As they may be derived, from any of the forms of vibration which 

 have been described, they may be circular or elliptical ; and they may be 

 performed in the tangent plane of the wave itself, instead along its radius. 

 In the ease of circular movements performed around the radius as an axis, 

 an undulation will then consist of a chain of particles occupying positions 

 less and less advanced, in the direction of progress, in Hieir respective circles, 

 until the last differs from the first by 3G0°. At any instant, therefore, the par- 

 ticles in such undulations, or series of undulations, will present the exact counter- 

 part, direct or reversed, of the thread of a screw. 



At a very great distance from the centre of disturbance the curved front of 

 the v/ave will become sensibly plane. If a solid obstruction be ])laced in the 

 way of such a plane wave, having in it a small aperture through which a minute 

 elementary portii;u of the movement may be propagated, while all the remain- 

 ing part is ari'estcd, then this aperture will become the centre of a new spherical 

 wave on the other side of the obstruction. If the solid be perforated by a great 

 number of apertures, each of these will generate its own independent wave; 

 and all these waves, as, in their progress, they encounter each other and become 

 blended, will ultimately reproduce anew the plane wave from which they origi- 

 nated. 13y supposing th(! number of these apertures infinitely great, and tlic 

 spaces between them infinitely small, we shall arrive at the conclusion that a plane 

 ivave is equivalent to an infinite, number of spherical waves, whose centres are 

 infinitely near to each other in that plane, and of Avhich the plane wave is the 

 resultant; so that if, at any time, we intercept any number of these component 

 waves, either such as are contiguous to each other, or such as are separated by 

 determinate intervals, the consequences of the proceeding may be calculated a 

 jyriori by finding the resultant of those whicji remain unobstructed. This princi- 

 ple, which is of the highest importance to the physical theory of undulation as 

 applied to optics, was first laid down by Iluyghens. Wc shall presently see 

 how it was applied by him to explain the phenomena of the reflection and re- 

 fraction of light. 



% 3. EEFLECTION AND HE FRACTION. 



If light is to be regarded as an effect of undulation, the elastic fluid in 

 which its undulations are propagated must be inconceivably more rare than the 

 air. This fluid must be supposed not only to fill all sjjace, but also to occupy 



