276 DR L. N. G. FILON ON THE 



(/! ju or )-gradient is of the order 10~ 4 . Taking ^ = f , the curvature is of order :-j 1 0~*. 

 Hence the greatest deviation from the straight line = (curvature) (thickness of 

 slab)' = 3 x 10~* a divergence which cannot possibly affect the results. 



Thus we are justified in treating the paths of the rays as linear. Moreover, the 

 divergence of the ordinary and extraordinary rays after refraction at entrance 

 = fji~ a (fjL^ fi or ) (angle of incidence) very nearly, and it is easily verified that the effect 

 of this is also entirely negligible. Therefore we may treat the two rays as geometrically 

 coincident. 



The paths of the rays being linear, the planes of polarization are fixed throughout. 

 For these can be proved to be the plane through the ray, and the line of strain and 

 the plane through the ray perpendicular to the first plane. And the line of strain is 

 always parallel to the axis of the slab. 



Also if ft = angle between ray and line of strain, the relative retardation introduced 

 by an element ds of path is 



Hence the total relative retardation is 



R = 2aCT siri a sec r , 

 where 



T = tension at mid-point of path, 



2a = thickness of slab, 

 y = inclination of ray to the horizontal perpendicular to the axis. 



In practice the limits for cos ft are + 0'01, and for y are 0'02. It follows that the 

 factor sin 3 ft sec y introduces a proportional error less than 10~ 4 in the relative 

 retardation. It may therefore be altogether neglected. 



8. Combined Effect of Flexure and Obliquity. 



The relative vertical displacement of the two slabs due to flexure varies with the 

 cross-section taken. Now the pencil of rays used passes through a comparatively 

 large region of the beams, extending to about 1 centim. on either side of the central 

 cross-section. It is readily shown that the changes in z lt z 2 , due to flexure as we 

 pass from the central cross-section to sections distant x from the central one are 

 given by 



82, = - Sc.o'W/ielWi* = -3W/E] 



Mf x=\, 

 VV = +3W/EJ 



the slab N being bent concave downwards and F concave upwards. 



The greatest possible change in Sz l Sz 2 , due to this cause, is numerically equal to 

 <;\\ r /E or (taking E = 600,000 kilogs.- weight per square centimetre) = Wx 10~ 6 . 



The proportional correction in the stress amounts to 10~ 5 W/^ 2.,) nearly, i.e., to 



