DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 271 



the slab the effect of such local perturbations must be negligible. (See ' PhiL Trans.,' 

 A, voL 201, pp. 114, 145.) 



When the apparatus is in perfect adjustment K is exactly midway (measured 

 horizontally) between U and V, and the horizontal distances between the edges of V 

 and R and of U and S are equal. If these be each o, then the bending moment 

 applied to the part of the slab between R and S is constant and equal to cW. 



For, since the reaction at the upper edge of V is vertical and the load at K is 

 vertical, then the reaction at the lower edge of U is shown to be vertical by 

 considering U, I, K as one system. Thus the reactions at the lower edge of U and V 

 are each equal to W. Again, the reaction at the upper edge of R is vertical and 

 therefore the reaction at the upper edge of S is also vertical. Hence these reactions 

 also are equal to W. 



Also it is to be noted that, if the adjustment be perfect, the bending moment 

 applied to the beam or slab is a pure bending moment. There is no total shear across 

 any cross-section between R and S. 



In such a case it is well known that the distribution of stress obeys accurately the 

 Euler-Bernouilli laws and consists only of a tension My/A/, J parallel to the axis of the 

 beam, where M = applied bending moment, y = height above neutral axis (horizontal 

 line drawn through the centroid of the cross-section in the plane of the cross-section), 

 A = area of cross-section, k = its radius of gyration about neutral axis. The formulae 

 (4) and (7) are therefore verified. 



In fig. 3 the knife-edges U and V are outside R and S. The bending moment is 

 therefore positive, with the convention of p. 266. For the second beam the arrange- 

 ment is the same, except that now U and V are inside R and S, so that the bending 

 moment is negative. 



The difference of height between the slabs was obtained by placing the knife-edges 

 R, S for one of the beams upon two steel blocks of height 0'5 centim. instead of 

 directly upon the bed-plate. The bed-plate itself was a solid plate of steel, very 

 strong and resting upon two heavy tables T of the same height. 



In the alwve description no account has been taken of a large number of small 

 errors which must theoretically affect the method. 



The principal are the following: (1) In the theory explained in 2 modifications 

 will be introduced owing to the fact that a polarizing Nicol is introduced in the path 

 of the rays of light lietween the source and the slabs. (2) The source of light is not 

 a line-source, but a slit of finite breadth. (3) When the load is applied, the middle 

 part of one beam rises and the other sinks: thus the heights z t , z, and the relative 

 height 2, 2, in formula (7) are not fixed. (4) The bed-plate P and the tables T are 

 nut absolutely rigid. This will alter 2, and 2,, but not 212* (5) The rays do not go 

 through the glass horizontally and at right angles to the axes of the slabs, and the 

 assumption that the mean retardation = retardation at mid- point of path is only an 

 approximation. (6) The slit used as a source of light is not accurately horizontal. 



