66 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



This quantity is small compared with 8,y, when the depth of the beam is 

 small compared with its length. 



The whole deflection &y = S,y + Sy 



(69). 



CASE XIII. 



Wlien the values of the compressions at any point have been found, when 

 two different seta of forces act on a solid separately, the compressions, when 

 the forces act at the same time, may be found by the composition of com- 

 pressions, because the small compressions are independent of one another. 



It appears from Case L, that if a cylinder be twisted as there described, 

 the compressions will be inversely proportional to the square of the distance 

 from the centre. 



If two cylindric surfaces, whose axes are perpendicular to the plane of an 

 indefinite elastic plate, be equally twisted in the same direction, the resultant 

 compression in any direction may be found by adding the compression due to 

 each resolved in that direction. 



The result of this operation may be thus stated geometrically. Let A l and 

 A t (Fig. 1) be the centres of the twisted cylinders. Join A^A t , and bisect A t A t 

 in O. Draw OBC at right angles, and cut off OB l and OB a each equal to OA,. 



Then the difference of the retardation of oppositely polarized rays of light 

 passing perpendicularly through any point of the plane varies directly as the 

 product of its distances from B l and B 3 , and inversely as the square of the 

 product of its distances from A l and A t . 



The isochromatic lines are represented in the figure. 



The retardation is infinite at the points .4, and A t ; it vanishes at , 

 and B, ; and if the retardation at O be taken for unity, the isochromatic curves 

 2, 4, surround A 1 and A s ; that in which the retardation is unity has two 

 loops, and passes through 0; the curves , are continuous, and have points 

 of contrary flexure ; the curve has multiple points at C l and (7,, where 



