654< Mr. stokes, on THE FORMATION OF THE CENTRAL SPOT 



23. Suppose the incident light to be polarized in a plane making an angle a with the plane 

 of incidence. Then at the point of contact the light, being transmitted as if the first and third 

 media formed one uninterrupted medium, will be plane polarized, the plane of polarization being the 

 same as at first. At a sufficient distance from the point of contact there is no sensible quantity of 

 lio-ht transmitted. At intermediate distances the transmitted light is in general elliptically polarized, 

 since it follows from (8) and the expression thence derived by writing (p for d that the two streams 

 of light, polarized in and perpendicularly to the plane of incidence respectively, into which the inci- 

 dent lio-ht may be conceived to be decomposed, are unequally accelerated or retarded. At the point 

 of contact, where q=l, these two expressions agree in giving •>/', = 0. Suppose now that the trans- 

 mitted light is analyzed, so as to extinguish the light which passes through close to the point of 

 contact. Then the centre of the spot will be dark, and beyond a certain distance all round there 

 will be darkness, because no sensible quantity of light was incident on the analyzer ; but at inter- 

 mediate distances a portion of the light incident on the analyzer will be visible. Consequently the 

 appearance will be that of a luminous ring with a perfectly dark centre. 



24. Let the coefficient of vibration in the incident light be taken for unity ; then the incident 

 vibration may be resolved into two, whose coefficients are cosa, sin a, belonging to light polarized 

 in and perpendicularly to the plane of incidence respectively. The phases of vibration will be 

 accelerated by the angles \^^, \|/^^, and the coefficients of vibration will be multiplied by p^, p^^, if \|/, , 

 o are what v|/^, jO, in Art. (l3) become when (p is put for 6. Hence we may take 



{27r , ] 



— {vt - hoc) + \|/ V , 



,|^(«<-Mr')+^|'„} 



sm a . cos • 



to represent the vibrations which compounded together make up the transmitted light, a' being mea- 

 sured in the direction of propagation. The light being analyzed in the way above mentioned, it is 

 only the resolved parts of these vibrations in a direction perpendicular to that of the vibrations in 

 the incident light which are preserved. We thus get, to express the vibration with which we are 

 concerned, 



sin a cos a |p,cos i'^ {vt - fix) + \l/\ - p cos [ — (u < - yu*') + '^„] \ , 



which gives for the intensity (/) at any point of the ring 



/ = isin-2a{(,o, cos x// - jo,,cos'>//^,)*+ (,0, sin\^^ - jo^^sin \//J'} ... (14), 

 = 1 sin=2a |p/ + p^; - 2p^ p^^ cos (x/,„ -i^)\. 

 Let Pg, Qg be respectively the real part of the expression at the second side of (7) and the 

 coefficient of \/- 1, and let P^, Q^ be what Pg, Q, become when <p is put for 9. Then we may 

 if we please replace (14) by 



I = is\n'2a\{P,-P^y+{Q,-QJl (15). 



The ring is brightest, for a given angle of incidence, when a = 45°. When i = i, the two kinds 

 of polarized light are transmitted in the same proportion ; but it does not therefore follow that the 

 ring vanishes, inasmuch as the change of phase is different in the two cases. In fact, in this case 

 the angles cp, 6 arc complementary ; so that cot 2rf), cot 26 are equal in magnitude but opposite in 

 sign, and therefore from (8) the phase in the one case is accelerated and in the other case retarded 

 by the angle 



tan-fi-HiJcotse), or tan-(i^^:^ 



