8 The Double Refraction of Light in a Crystallized. 



in the direction of s let them "be a", Z>", c". Then, if a point 

 receive an equal displacement in a direction 01 making with 

 OX, OY, OZ, the angles a, j3, 7, the forces in the direction of 

 x t y, z (denoting then by X, Y, Z, respectively), will be 



X = a cos a + a' cos j3 + a" cos 7, 

 F = b cos a + V cos ft + V cos 7, 



Z = C COS a + C COS ft + G' COS 7, 



as follows from considering (see Fresnel's Memoir, p. 82) that 

 the force arising from a displacement in any direction is the 

 resultant of the forces arising from the three displacements in 

 the directions of x, y, z, which are the statical components of 

 that displacement. But since (p. 90) the force in the direction 

 of one of the axes, arising from a displacement in the direction 

 of another, is equal to the force in the direction of the latter, 

 arising from an equal displacement in the direction of the former, 

 it follows that a' = b, a" = c, b" = c' ; and hence 



X = a cos a + b cos ft + c cos 7, 

 T = b cos a 4 b' cos ft + c f cos 7, 

 Z = c cos a + c' cos ft + c" cos 7. 



Let ax 1 + b'y 1 + c"s 2 + 2c'yz + 2czx + %bxy - 1 be the equation 

 of a surface of the second order. Let 01 intersect it in /, the 

 co-ordinates of / being x', i/, z' ; then the equation of the tan- 

 gent plane at / will be, by the known formulae 



(ax' + by + cz'} x + (bx' + b'y' + cz'} y + (ex' + cy' + c"z') z = 1. 



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

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



-^p = ax' + by + cz' = r (a cos a + b cos ft + c cos 7) = rX, 



= bx + b'y +cz' = r (b cos a + b' cos ft + c cos 7) = rY, 



OQ 



= ex + c')/ + c"z' = r (c cos a + c cos ft + c" cos 7) = rZ. 



