COLORS OF THIN PLATES. 185 



in the two conflicting rays ought to bo equal in order to produce this cff:;ct. Now, 

 if we af-Humo (what Avill hereafter be proved) that llie amount of h'glit reflected 

 at cither surface is in a constant ratio to the amount of light incident upon it, 

 when the angle of incidence and the index of refraction are themselves Constant, 

 we shall perceive that the ray which emerges after one reflection at the lower 

 surface is feebler than that which is reflected at the upper: for the light incident 

 upon the lower is already enfeebled by the loss at the upper, and the reflected 

 ray is again diminished by the second reflection which occurs at its emergence 

 through the upper. But the light which is thus turned back at the upper surface 

 is again reflected at the lower, and at its return another portion emerges through 

 the upper. A series of reflections thus goes on between the two surfaces, each 

 one contributing to strengthen the emergent ray ; and the resultant of all these 

 contributions is to bring the ray from the lower surface, in the end, up to exact 

 equality with that which is originally reflected from the upper without entering 

 the lamina. This will appear to be rigidly true if we consider the following 

 statement. The intensity of light is measured by the living force which ani- 

 mates the mass of ether in which the molecular movements are going on. Let 

 the masses in the two adjacent strata of the two media v/hich act upon each 

 other be distinguished by the letters m for the denser and m' for the rarer. Now 

 it is true (as v/ill hereafter be proved) that the velocity of molecular movement 

 in a wave reflected from the separating surface of two given media, at a given 

 incidence, bears a detenninate ratio to the molecular velocity in the incident wave. 

 Let this be represented by th(; ratio v \ 1, the incident velocity being unity. 

 Then the living force of the reflected wave will be viv^, and that which passes 

 into the other mc^dium and forms the transmitted wave will be m{\ — r^). Ac- 



on 

 cordingly -/l — v") is the square of the molecular velocity in the transmitted 



w^avc. Let it be represented by i^. 



By reflection at the second surface, u becomes vu, and this, by another rc-flec- 

 tion of the returning wave at the first surfoce, becomes v^u. From tlie living 

 force of the wave returning from the second surface subtract the living force 

 which it loses by the second reflection at the first, and the remainder, which is 

 the living force transmitted through the first surface, will be vi' 6h^ {\.-~i^). 



And this, divided b}' the mass m, gives — v^u^ (l—v^) for the square of the first 



m 



component of the molecular velocity in the wave which reaches the eye from the 



second surfiice. In like manner v^u becomes v''u by second reflection at the 



second surface ; and v^u becomes vf^u by the succeeding reflection at the first 



surface. And the expression for the square of the second component of the 



molecular velocity we are seeking, vrill be — ifii" (1 — t^). The next term will 

 m' ,/ m ^ ' ■ 



ho — v^'^u^ {\~xr)\ and from a comparison of these three the law is evident. 

 m 



By substituting the value of a^, taking the square roots of these sqiiares and 



making their sura, which is the resultant molecular velocity of the vrave emergent 



from the second surface, equal to i^ , we shall obtain — 



-2; 



The sum of the series in parentheses is —^ . Hence t»'=?;, or the reflected 



v'' — 1 

 velocities, and consequently the intensities, of the waves reflected from the 

 two surfaces, are equal. 



It is assumed in the foregoing that all these components agree in phase. But 

 this is evidently true at the points where the dark rings, as seen by reflection, 

 appear. For at these points the first return wave from the lower surface is in 



