INTENSITY OF REFLECTED AND REFRACTED LIGHT. 397 



per signification of density within a medium, nor, if we had, is it quite obvious 

 that the aggregate attraction of combined molecules should of necessity vary as 

 the density. It must be recollected that, in estimating the effect of forces result- 

 ing from molecules, the whole result consists of the sum of a large number of 

 terms, not diminishing in magnitude with the same rapidity in all cases. It does 

 not then follow, that the attraction varies as the density, nor even according to 

 any simple function of it. 



But I do not stop here. Allowing the assumptions of Fresnel to be correct, 

 — and from the coincidence of Mr Green's conclusions with his, most persons will 

 be inclined to think them substantially so, — all the discrepancy between the 

 molecular hypothesis as viewed by M. Cauchy, and that deviation from it adopt- 

 ed by M. PoissoN and Mr Green, amounts to this, that one party (suppose the 

 former) have misinterpreted the formulae relative to the density of the particles. 

 I shall shew presently that the formulae themselves are not at all affected by the 

 apparent contradiction of conclusion, since the results of M. Fresnel may be 

 deduced as easily, and I think with as little assumption, by the molecular hypo- 

 thesis as by the other. 



ANALYTICAL INVESTIGATION. 



My object in the investigation which follows, is to deduce M. Fresnel's for- 

 mulae for the intensity of rays reflected at the common surface of two media, air 

 and glass, the incident rays being polarized. It will not be requisite in this place 

 to enter into a discussion of the results obtained by grouping particles. Suffice 

 it to say, that, by strict mathematical investigation, it can be shewn that the as- 

 sumption of Newton's law of force for the particles of the media surrounding the 

 material particles, gives rise to an expression of the following form, for the ag- 

 gregate attraction or repulsion of those particles which surround one particle of 

 matter 



— aa l + aa 

 f = m.e 



a" 



a being the distance between two material particles. This expression is insensi- 

 ble at sensible distances, and consequently we may limit our summation in the 

 subsequent process to such distances. We will adopt the following notation. 



The media being both perfectly symmetrical, and bounded by a plane sur- 

 face ; let that plane be called the plane of p z, the axis of z being parallel to the 

 line at which the front of the wave cuts the plane, and that of of the direction of 

 transmission. When the incident vibrations are polarized in the plane of inci- 

 dence, all the motion will be in a direction parallel to the axis of z. 



Take x, y, z as the co-ordinates of any particle in a state of rest ; x,y,z+'y 



vol. VIV. PART II. 81 



