REPORT ON PHYSICAL OPTICS. 309 



principle of the coexistence of small motions. It easily follows 

 from this mode of composition, tliat the sm'face of the primary 

 wave must mark the extreme limits to which the vibratory move- 

 ment is propagated in any direction, in any given time ; so that 

 light, according to this theory, is propagated from any one point 

 to another in the least possible time. This is the well-known 

 law of Fermat, the law ofstviftest propagation, and it Avill rea- 

 dily appear that it holds, whatever be the number of modifica- 

 tions which the course of the light may undergo by reflexion or 

 refraction ; as, likewise, whatever be the form of the elemen- 

 tary wave. 



The demonstration of Huygens has been thrown into an 

 analytical form by Lagrange*, but he has added nothing to its 

 rigour or perspicuity. An important supplement to the demon- 

 stration was however given by Fresnel. From the reasoning of 

 Huygens it did not appear what became of those portions of 

 the secondary waves which did not conspire in the formation of 

 the grand wave. The crossing of these in all directions ought 

 to give rise to a weak diffused light, filling the entire space be- 

 tween the grand wave and the reflecting or refracting surface ; 

 and, in fact, Huygens supposed that such a light did actually 

 exist, but was too feeble to affect the eye. Fresnel has shown, 

 however, that all those portions which do not conspire in the 

 formation of the grand wave, are destroyed by interference f ; 

 so that the formation of one grand wave, by the union of an in- 

 definite number of lesser waves, becomes a precise and definite 

 effect. 



The total reflexion of light at the surface of a rarer medium 

 has been urged by Newton against the wave-theory, and the 

 apparent difficulty seems to have had much weight in inducing 

 him to reject that theory. It is, in fact, not easy to perceive at 

 first view wl y the disturbance of the ether within the denser 

 medium shoi d not be communicated to the external ether, and 

 a wave be thus propagated to the eye, whatever be the obliquity 

 of the incident wave. To this it may be enough to reply, that 

 the law of refraction itself, in all its generality, is a necessary 

 consequence of the wave- theory ; and therefore that the phe- 

 nomenon of total reflexion, which is a particular case of that 

 law, is likewise accounted for. But the principle of interfer- 

 ence furnishes a direct answer to the difficulty. It can be 

 shown that the elementary waves, which are propagated into the 

 rarer medium from the several points of the bounding surface, 



* " Sur la Theovie dc la Lumiere d'Huygens," Annales de Chim., torn, xxi, 

 t 'I Explication de la Ilefractlon dans la Systeniu dcs Ondcs," Annates de 

 Chimie, torn. xxi. 



