402 PROFESSOR KELLAND ON FRESNEL'S FORMULAE FOR THE 



of surprise, even supposing in both cases the argument fallacious ; for in all the 

 ways of establishing them the same grand assumption extends throughout the 

 whole, viz. that the particles at the common surface of the media have motions re- 

 sulting from, and conversely affecting, the motion without the latter medium, and 

 that these motions are regulated by the usual laws of the result of forces. Per- 

 haps I shall be better understood if I illustrate my meaning by giving the follow- 

 ing demonstration of the results in question. 



A particle at the surface is acted on by three sets of forces, in the directions 

 respectively of the directions of incidence, reflection, and refraction : not that the 

 particle is urged in these directions, but is acted by a force which gives it a mo- 

 tion as much depending on the direction as though it were. We have then three 

 forces acting on the particle, and any one may be considered as the resultant of 

 the other two. If this be allowed, we know by the laws of mechanics, that each 

 force is in the proportion of the sine of the angle contained by the other two. 



Let then I R and T denote the incident reflected and transmitted vibration ; 



then ^ = ^^^^t^ 



A svnip + (p, 



T sin 2 (^ 



I sin ^ + (p^ 



2 sin cos (p 



sin (p + <p, 

 2 fx sin <p^ cos (p 

 sin (p + (p, 

 fj. being the refractive index. 



But when the motion actually takes place within the medium, the length of 



the wave has to be diminished in the ratio - : 1 ; if, then, we conceive the new 



wave to remain similar to the old one, as we doubtless ought, we must diminish 

 the vibration in the same ratio : hence the value of the vibration within the me- 

 dium is 



1 rp 1 J 2 sin cos (p 



M M sin <p + (p, 



J 2 sin (p,cos<p 

 sm(p-t (f), 

 the same result as before. 



This consideration, then, leads us to M. Fresnel's formulae. 



6. Next let us adopt the molecular hypothesis, without having recourse to 

 the approximations mentioned in the introduction : then 



df 



= j(^.,+ ^a-)a7. 



