30 REFLEXION AND REFRACTION. 



in the new medium, and with the velocity which belongs to 

 it ; so that there will be an infinite number of partial wares in 

 both media, diverging from the several points of the bounding 

 surface. But, by the principle of the coexistence of small 

 motions, the agitation of any particle of either medium is the 

 sum of the agitations sent there at the same instant from these 

 several centres of disturbance. The surfaces on which these 

 are accumulated will be the reflected and refracted waves, and 

 they are obviously those which touch all the small spherical 

 waves at any instant. 



Thus, let mn be the front of 

 a plane wave, meeting the re- 

 flecting surface at m. Each 

 portion of this wave, as it 

 reaches the surface, becomes 

 the centre of a diverging sphe- 

 rical wave in the first medium, which will be propagated with 

 the velocity of the original wave. Accordingly, when the 

 portion n reaches the surface at k, the portion m will have 

 diverged into the spherical wave, whose radius, mo, is equal 

 to nk. And, in like manner, if m'n 1 be drawn parallel to 

 mn, the wave diverging from m will in the same time have 

 reached the spherical surface whose radius, m'o', is equal to 

 n'k. The surface which touches all these spheres at any 

 instant is that of the reflected wave. But, as mo and m'o' are 

 proportional to mk and m'k, it is obvious that this tangent 

 surface is plane ; and since mo = nk, and the angles at n and 

 o are right, it follows that the angles nmk and okm are equal, 

 or that the incident and reflected waves are equally in- 

 clined to the reflecting surface. 



(37) The proof of the law of refraction is in all respects 

 analogous to the preceding. Let mn be the position of the 

 incident plane wave at any moment. In an interval of 



