POLARIZATION BY REFLECTION. 101 



Let us next examine the case of a wave pMlarizcd in a plane at riglit angles 

 to the ])lane of incidence. In this case, the mok^cular movements arc- in i\w. 

 plane of incidence. The expressions for the masses will be the same as before ; 

 but the components of the molecular velocities 'parallel to the rrflccti-ng sniface 

 are to be taken, instead of the velocities themselves. These components are, 

 for the incident Avavc, IX cos:; for the reflected wave, ^''Xcos;; and for the trans- 

 mitted wave, tc' Xco» fi : and the assumption of Fresnel is — 



(1 + ?/)cos: = u'coBp* 



* In the attempt to apply the theory of iindulatiou to th(^ case of reflection and refraction 

 at the surfaces of crystalline media, it has been found more satisfactory by Profs. McCnlla<Th 

 and Neumau to employ the following assumptions, viz: 



I. The vibrations of polarized light are ])arallel to the plane of po'!arization. 



II. The density of the ether in both media is the same as in vacuo. 



III. The vis viva 'is preserved. 



IV. The resultani of the vibrations is the same in both media; and therefore, in singly 

 refracting media, the vibration of the refracted ray is the resultant of the vibrations of the 

 incident and reflected rays. 



On these principles, the case of reflection in the plane of polarization is simp'e. Let the 

 refracted ray be extended backward, and it will divide the angle between the incident and 

 reflected raj's (which equals 20 into two parts, which are, respectivcij', i-\-p and t — p. Upon 

 this retrbduccd line as a diagonal, if a parallelogram be constructed with the incident and 

 reflected rays as sides, this diagonal and these sides will be proportional to the amplitudes, 

 and therefore to the velocities, of the molecular movements which are perpcmdicular to them 

 severally. Employing then, as before, unity and the symbols v and u to designate these 

 velocities, we shall have directly — 



sin(t — p) sin'2i 



v^= — . M= . 



s'm{c-{-p) siu(i+p) . 



In the case of reflection in a plane at right angles to the plane of pohirizatiou the vibra- 

 tions are all parallel, and the fourth principle above gives — 



I-f-»'=i<'. 



Also the third principle gives (m and m' repr(^senting the masses of the ether put into mo- 

 tion in the contiguous strata of the two media) — 



?)t(i — v'-)=>n'it''^. 

 These equations lead to the values — 



, ?« — m' 2m 



m-\-)ii' m-\-7u' 



The same values may also be deduced from the laws of impact of elastic bodies 



According to the second principle, the masses m and to' are proportional to their volumes ; 

 and these volumes have been found in the text to be proportional to sm^cosi and sinpco.s/i. 

 Substituting these expie-ssions in place of m and m' in the foregoing iiuctiony, we .shall 

 obtain — 



, tan(f — p) I sin'2i ^_^ 2sin2i 



tan(t-|-p)' sin,^i-t-pjcos(i— p) tan^i-|-p)i_cos--'i-f cosJJp)' 



_ Comparing these values with those of ihe text,' we tind those or v alike, but witli reversed 

 Signs and interchanged— that which repiesented the velocity of molecular movement mor- 

 vial to the p'ane of po'arization before, now denoting the velocity of movement in the 

 plane, and v. v. In the expressions for the transmitted rays there is a difierence which 

 i-e.-Tilts from the adoption of the second of the principles fo;egoiug, making the density of 

 I he ether in the two media the same, which is not the supposition of the textr 



However strongly on some accounts this view of the subject may seem to recommend 

 itself to our acceptance, it introduces a difficulty (elsewhere noticed) into the theory of double 

 refraction, which has never j-et been met, and which seems to have been sino-ularly ip-nored 

 by many who have engaged in this discussion. ° ° 



In order to facilitate the comparison of the values of the several expressions fore^-oin"" 

 they may be reduced to a simple form of common denominator, when they become— ^ ° 



1. For the case of vibration in the plane of reflection — 



_ sm2{c — p) _ 2sin0tcos(i — p) 



sin'2i-}-siu2p" siu^it+siuiJp 



2. For vibration normal to the plane of reflection — 



, sin2t — sin2p , 2sin2i 



r= -. tt'= . — 



sm2i-)-sin2o sin2£-f-sin2p 



