73 



Put 01 = r, and let the tangent plane intersect OX, OY, OZ, in 

 the points P, Q, R ; then from this equation we have 



y-T^ = ax' + by' -\- cz' z= r (a cos. « -j- 6 cos. /3 + c cos. y) = rX, 

 z: l>x' -\- b'y' -\- c'z' ^r {b cos. »-{- b' cos. /i + c' cos. y) r: rF, 



0« 



1 

 Uli 



— — = c.t' -}- c'y' -{- c"z' = »• (c cos. « -f- c' cos, /3 ^ c" cos. y) = rZ. 



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

 tangent plane, the cosines of the angles which this perpendicular 

 makes with the axes of coordinates will be equal to 



P p p 



OP' OQ' 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^/ X'+ Y' + Z'zzl. 



Hence it appears that the perpendicular let fall from O on the 

 tangent plane is parallel to the direction of the resultant elastic force, 



and that the magnitude of the resultant is expressed by ~- 



From this conclusion we may, with the greatest facility, deduce 

 several corollaries. 



1", Since the elastic force is supposed finite, whatever be the direc- 

 tion of the displacement, it is manifest that the above-mentioned sur- 

 face of the second order must be an ellipsoid, and that when the dis- 

 placement 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 elas- 

 ticity at right angles to each other. —(Memoir, p. 93.) Also the 



VOL. XVI. M 



