FRESNEL ON DOUBLE REFRACTION. 287 



Substituting these two values in the first differential equation, 

 which expresses the common condition for a maximum or for a 

 minimum, we find for the equation determining the dii'ection of 

 the axes of the diametral section, 



a2cosX(BcosZ-CcosY) + 62cosY(CcosX + cosZ) 



- c^ . cos Z (B cos X + cos Y) = (A.) 



Let us now conceive a plane drawn through the radius vector, 

 and the accelerating force developed by the displacements parallel 

 to the radius vector ; it is in this plane that we shall decompose 

 this force into two others, the former in the direction of the 

 radius vector, the second perpendicular to it ; and if this plane 

 is perpendicular to the cutting plane, it is clear that the second 

 component will be normal to this latter. We proceed now to 

 find the equation which expresses that these two planes are at 

 right angles to each other ; and if it agrees with equation (A.), 

 we may conclude from it that the axes of the diametral section 

 are precisely the two directions which satisfy the condition that 

 the component perpendicular to the radius vector be at the same 

 time perpendicular to the cutting plane. 



Let X = B'y + C'zhe the equation of the plane drawn through 

 the radius vector, and the direction of the elastic force developed 

 by the vibrations parallel to the radius vector. The cosines of 

 the angles made by this force with the three axes of coordinates 

 are 



ft^cosX b'^.cosY c'^.cosZ 



^7^' ~~J^' ~7^' 



and since it is contained in the plane a.' = B' Y + C Z, we have 



fl^.cosX Ti, i^.cosY „, c^.cosZ 

 ^^=B'.^+e.-^, or 



«2 . COS X = B' . *^ cos Y + C . c2 . cos Z. 

 This plane containing the radius vector, we have, similarly, 



cosX = B'.cosY + C'.cosZ. 

 From these two equations we obtain 



_ (a^-c^)cosX _ (a^-6^)cosX 



^ - (62_c^)cosY """"^ ^ " {b^- O cosZ- 

 Substituting these values of B' and C in the equation 



B B' + C C + 1 = 0, 

 which expresses that the second plane is perpendicular to the 

 first, we find 



