Intelligence and Miscellaneous Articles. 397 



ON THE REFLECTION OF POLARIZED LIGHT. 

 BY M. CROULLEBOIS. 



Among the fringes the discovery of which we owe to Mr. Airy- 

 there is one, called courbe en semelle by M. Billet, remarkable for 

 the instability of its form and orientation, and therefore very ap- 

 propriate to serve as a characteristic. I purpose to show the 

 advantage that may be derived from the study of this curve in 

 order to ascertain : — -1, the physical constitution of a mirror (that 

 is to say, its positive, neutral, or negative nature) ; 2, the value of 

 the angle of maximum polarization (first constant) ; 3, the azimuth, 

 of restored polarization (second constant). 



To obtain, this fringe Jamin's apparatus is used, in the ordinary 

 manner, but substituting for the compensator a convergent system 

 formed by a combination of a lens and a perpendicular spar of 

 several minims, thickness. It is, besides, advisable to make use of 

 the homogeneous light of salted alcohol. The mirror giving all the 

 states of ellipticity, the calculation of the isochromatic curves must 

 be effected in a very general case. 



Suppose two axes traced in a plane perpendicular to the reflected 

 ray: — the one horizontal, OX, lying in the first azimuth (the plane 

 of incidence); the other vertical, OT. Let OP = l be the initial 

 vibration, having its direction in the upper right-hand quadrant ; 

 and let a, (3, y be the angles formed respectively with OX by the 

 vibration OP, by the principal section of the plate, and by that of 

 the polariscope. The definitive extraordinary image is formed of 

 four components, of which the amplitudes and phases are : — 



A= cos <r cos (3 cos (y— /3), 0+E ; 

 B= sin er sin /3 cos (y — (3), ^' + E; 

 C= sin o- cos /3 sin (y— (3), $' + ; 

 D=— cos a sin (3 sin(y— /3), + 0. 



<p and (f>' designate the phases introduced by reflection, O and E 

 those of the two rays in the spar. In addition, we have put 



li cos a = cos (t, k sin a = sin <r. 



To calculate the intensity I of the image, we further put 



0— 0'=a, 0-E=2', 



and we find the following value — 



I=(2Acos£) 2 + (2Asin£) 2 . 



The isochromatic curves are given by the equation 



which, rendered explicit, gives 



,v — sin' 



tan 8 = — r— — 



cos 2<r sin 2/3 — s 



jo being proportional to the distance from the centre of the field to 



, v — sin 2cr sin 3 , „ 



tan o = = tan 2iro~, 



cos 2<r sin 2(3 — sin 2a cos 2(3 cos & 



