Medium , according to the Principles of Fresnel. 9 



Now if p be the length of the perpendicular let fall from 

 on the tangent plane, the cosines of the angles which this per- 

 pendicular makes with the axes of co-ordinates will be equal to 



P P P 

 OP OQ 1 OR' 



respectively, that is, to prX, prY, prZ\ and since the sum of the 

 squares of these cosines is equal to unity, we have 



pr v 2 + F 2 + Z z = 1. 



Hence it appears that the perpendicular let fall from on 

 the tangent plane is parallel to the direction of the resultant 

 elastic force, and that the magnitude of the resultant is ex- 



pressed by . 

 J pr 



From this conclusion we may, with the greatest facility, 

 deduce several corollaries. 



1. Since the elastic force is supposed finite, whatever be the 

 direction of the displacement, it is manifest that the above-men- 

 tioned surface of the second order must be an ellipsoid, and that 

 when the displacement is in the direction of any of the three 

 axes of the ellipsoid, the elastic force excited will be in the 

 direction of the same axis, because the tangent plane will be 

 perpendicular to it. Hence the remarkable consequence, that 

 there are always three axes of elasticity at right angles to each 

 other. (Memoir, p. 93.) Also the 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 extre- 

 mity of that semidiameter ; and for equal displacements the 

 magnitude of the force will be inversely as the rectangle under 

 the semidiameter and the perpendicular from the centre on the 

 tangent plane at its extremity. If the displacements are pro- 



