300 FHESNEL ON DOUBLE REFRACTION. 



the light divides itself in traversing the crystal. Now it is evi- 

 dent that through the same straight line, and on the same side 

 of their common centre, there can only be drawn to them two 

 tangent planes ; for if three of these could be drawn, it would be 

 equally possible to draw three parallel tangent planes on the 

 same side of the centre of the waves, whence would result three 

 different distances of these tangent planes from the centre, and 

 consequently three velocities of propagation for the indefinite 

 plane waves parallel to one and the same plane, and we have 

 just shown that there cannot be more than two of these. For 

 the same reason there cannot be more than two points of contact, 

 for the existence of three points of contact would render possible 

 that of three parallel tangent planes. 



Calculation of the surface of the waves, continued. 



But in calculating the equation of the surface of the waves, 

 the degree of this equation will show us still more clearly that it 

 is impossible to draw to them, through one straight line, more 

 than two tangent planes on the same side of the centre. 



The equation of a plane passing through the centre of the 

 surface of elasticity being 



z = 7nx + ny, 

 that which determines the two values of the greatest and least 

 radius vector comprised in this diametral section is, as we have 

 seen, 



(e2_u2-) (c2_„2) „2 ^ (^,2_„2-) (c2_„2) ^2 



+ (a2-u2^ {b'^-v'') = (A.) 



We have already put for the equation of a plane parallel to the 

 section, 



the square of the distance of this plane from the origin of coor- 

 ds 

 dinates is represented by — — — 2~T — 2 5 hence to express that 



the plane parallel to the diametral section is distant from it by 

 a quantity equal to the greatest or least radius vector, it is suffi- 

 cient to write 



, , ^2'. 2 = o'. or C-^ = 0- (1 + »»' + n^). 



Hence the equation of this plane, to which the luminous wave 

 must be tangent, becomes 



[z — mx — nyY^ = V- [l -{- in- + n^) (B.) 



