POLARIZATION BY REFLECTION. 189 



that of the reflected wave, and u that of the transmitted wave, we sliall have 

 the equation, l-]rv^=u. 



With these suppositions let le, Ic be the; bounding 

 limits of a mass of the ether, along which an undulation 

 moves, meeting the reflecting surface; J\IN in cc. Let 

 f.'R, pR' be the limits of the reflected undtilation, and 

 rT, t'T' those of the transmitted undulatinn. Draw 

 <h, cf perpendicular to \c, tT. Let c(] be \\m length 

 of the incident, and eh that of the transmitted undu- 

 lation. Draw ad, gh parallel to ch, ef, respectively. 

 We may regard the incident undulation as a mass 

 whose bulk is the prism ahcAl, and density o, impinging upon a mass whose bulk 

 is cfgh, and density o' . Since the molecular movements in this case are in the 

 the common surface of the media, the breadths of the prisms, according to the 

 second assumption foregoing, are equal to each other. Their lengths will be k 

 and /', and their depths be and I'f. Or, putting m and m' for their masses, 

 m \m>:\ bcX^x'^ • tfx^' X^' Now, the wave lengths are proportional to the 

 velocities of progress of the wave in the two media, (which we may denote by 

 V and y.) Hence, if we put i for the angle of incidence, and p for the angle 

 of refraction, we shall have, A : A' :: V : V : : sin: : siu/3. Also, erbz=z'., and 

 • ef-=p; from which we derive Z»c:=6-c.cosj, and cf=ce.cos^p. And substituting — 



m : m' :: sin:cos:.(J : sinpcosp.d'. 



But, in the propagation of tremors through elastic media, the velocity of pro- 

 gress is directly as the square root of the elasticity, and inversely as the square 



root of the density. Or, putting £ for the elasticity, V^=-, and V'^=: ^,, £ be- 

 ing constant. Accordingly — 



^■' ^' '•' v2 * \rr2 ' ^^^' ^^y substitution, 



^ sin;cos£ ampco^p sin.'cosf smpcoBp cose cosjo 



V V sm'': siw^p sm: smp 



Now the living force in the reflected and transmitted waves must be equal to 

 the living force in the incident ; or — 



mxl"^?nxv^ + ?n'xu'^; that is, 



cose COS£ „ COBp „ /HO cos: COS/) . 



—. — = ——v^+ .~-u^; or, {\ — v'').'-r- =^ -r-^iif. 

 sm: szn: snxp sm: sm/> 



From this, with the equation previously given, «= 1 + y, we obtain, by elim- 

 ination, 



If, in order to embrace in the formula only the angle of incidence, we elimi- 

 nate p from the foregoing by means of the equation sin:=Ksin/), we shall obtain — 



(\/«2-sin2:— co3;\2 / 2cos: \2 . , . . , 



— — ^^ — ) , and M^— I . j . [:31.J ['2e.J 

 v«"^— sin^: + cos:/ \ v ra^— sint + cos:/ 



When the incidence is perpendicular, : = 0^, sin: = 0, and cos: = 1. lu this 



case, Z/'^=f — - B 5 ^^^ w^=| | . 



The intensity of light is measured by the living force of the molecular move- 

 ments, or by the mass multiplied by the square of the velocity. As the mas» 



