OF NEWTON'S RINGS BEYOND THE CRITICAL' ANGLE. 649 



Let the coefficient, which is imaginary, be put under the form p (cos \1/ + x/- 1 sin \l/) ; then 

 the real part of the whole expression, namely 



p cos [k (vt - Lv + nz) + \^|, 



will represent the vibration in the reflected light, so that p- is the intensity, and \l/ the acceleration 

 of phase. 



12. Let i be the angle of incidence on the first surface of the plate of air, /n the refractive 

 index of glass ; and let \ now denote tlie length of wave in air. Then in the expression for q 



, , Stt / , . , 



In the expression for b we must, according to Art. 2, take the imaginary expression for cos i' 



with the negative sign. We thus get for light polarized in the plane of incidence (Airy's Tract, 



p. 362, 2nd edition*), changing the sign of \/ - 1, 



6 = cos 20 + \/^l sin 2 6, 



where 



V// sin^j - 1 



tanO = -^ — (2). 



fi cos i 



Putting C for the coefficient in the expression (G), we have 



1 -q' 1-9' 



vhence 



I 



vhere 



6"' - q-b (1 - q') cos 20 - vA- 1 (1 + ?') sin 20 



_ (1 - <f) 1(1 - g") cos 20 + \/^ (1 + q") sin 20} 

 - fff + i-q' sin' 29 



1 + a' 



tanxf/ = \ tan 20 (3), 



' 1 — q^ 



e = (' - TT ... 



P (1 -q-)-+ iq'sin'2e ^ '' 



9 = e * (5). 



If we take p positive, as it will be convenient to do, we must take \l^ so that cos\l/ and cos 20 

 may have the same sign. Hence from (3) sin \// must be positive, since sin 2 is positive, inasmuch 



as lies between and — . Hence, of the two angles lying between - tt and tt which satisfy (2), 



we must take that which lies between and tt. 



For light polarized perpendicularly to the plane of incidence, we have merely to substitute rf) for 

 in the equations (3) and (t), where 



»o„^ //\/M'sin^J- 1 



tan© = (01. 



' cos I 



The value of q does not depend on the nature of the polarization. 



* Mr. Airy HpcakH of" vilirationn perpendicular to tlic plane of 1 which rc(|uireH us to enter into the (luculion whether the vibrntlonB 

 incidence," and " vibrations parallel to the plane of incidence," in jdane polarized li^llt are in ur perpendicular to the piano of 

 adopting the theory uf Fresncl ; but there is nothing in this paper I polarization. 



