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elasticities arising from equal displacements in the directions of the 

 three axes are inversely as the squares of the axes ; and hence, the 

 positions of the axes, and the elasticities in their respective directions 

 being given, the ellipsoid may be constructed. 



2°. The ellipsoid being thus constructed, the direction of the 

 elastic force, arising from a displacement in the direction of any of its 

 semidiameters, will be parallel to the normal at the extremity of that 

 semidiameter ; and for equal displacements the magnitude ot the 

 force will be inversely as the rectangle under the semidiameter and 

 the perpendicular from the centre on the tangent plane at its extre- 

 mity. If the displacements are proportional to the semidiameters, 

 the elastic forces will be both parallel and proportional to the nor- 

 mals ; for the normal, terminated by any of the principal planes, is 

 inversely as the perpendicular on the tangent plane. 



3°. If the elastic force be resolved into two, one parallel and the 

 other perpendicular to the direction of the displacement, the former 

 will be inversely as the square of the semidiameter in the direction of 

 the displacement. 



4". If the ellipsoid be cut by a plane through its centre, and if 

 the elastic force arising from a displacement in the direction of either 

 axis of the section be resolved parallel and perpendicular to that axis, 

 the part perpendicular to the axis will also be perpendicular to the 

 plane of the section. For (by lem, 4.) the plane passing through one 

 axis and the perpendicular to the tangent plane at its extremity, is 

 perpendicular to the other axis, and therefore to the plane of the 

 section. But if the displacement be in the direction of any other 

 diameter of the section, the elastic force, resolved perpendicularly to 

 that diameter, will be oblique to the plane of the section. 



To apply these things to the double refraction of light in a crys- 



