THE EQUILIBRIUM OF ELASTIC SOLIDS. 69 



of the action at any other point is determined by its position with respect to 

 the isochromatic curves. 



The direction of the principal axes of pressure at any point is found by 

 transmitting plane polarized light, and analysing it in the plane perpendicular 

 to that of polarization. The light is then restored in every part of the triangle, 

 except in those points at which one of the principal axes is parallel to the 

 plane of polarization. A dark band formed of all these points is seen, which 

 shifts its position as the triangle is turned round in its own plane. Fig. 3 

 represents these curves for every fifteenth degree of inclination. They correspond 

 to the lines of equal variation of the needle in a magnetic chart. 



From these curves others may be found which shall indicate, by their own 

 direction, the direction of the principal axes at any point. These curves of 

 direction of compression and dilatation are represented in Fig. 4 ; the curves 

 whose direction corresponds to that of compression are concave toward the 

 centre of the triangle, and intersect at right angles the curves of dilatation. 



Let the isochromatic lines in Fig. 2 be determined by the equation 



k(x,y) = I- = >(q-p)-, 



where / is the difference of retardation of the oppositely polarized rays, and 

 q and p the pressures in the principal axes at any point, z being the thick- 

 ness of the plate. 



Let the lines of equal inclination be determined by the equation 



bemg the angle of inclination of the principal axes ; then the differential 

 equation of the curves of direction of compression and dilatation (Fig. 4) is 



By considering any particle of the plate as a portion of a cylinder whose 

 axis passes through the centre of curvature of the curve of compression, we find 



' ' ' - = 



