lOS DOUBLE REFRACTION OF CRYSTALS. 



tion of the refracted ray. In fact it is easy to show, that in 

 this construction any given point o of the luminous wave 

 arrives at i, through the broken line oci in the same time as O 

 arrives at K. CI representing the velocity of the reOarted 

 ray, the right line C I is traversed in the same time as the 

 right line OK. Let us lake this for the unit of time, and O K 

 for the unit of space. The point o arrives at c in a time pro- 



C c 

 portionate to o c, and consequently equal to r-—. It passes from 



c to i in the interior of the crystal, in a time equal to that 



which the light ercploys in passing from C to I multiplied by 



_^, and consequently equal to— -i' c i being parallel to C I. 

 C K. K C 



By adding this time to —1 we shall have unity for the lime 



that the point o employs in arriving atz. 



Let us take o c infinitely near to o c, and parallel to it, the 

 point will arrive at i in the unit of time. Draw the right 

 lines CO and c i, and suppose, that the point o proceeds to i 

 through the broken line o c i. Now c o being perpendicular 

 to C O, the right line c o may be supposed equal to c o, and 

 the times required to pass through them may be supposed equal. 

 Moreover, the time required to pass through c i may be sup- 

 posed equal to the time required to pass through c i, because, 

 the plane K I touching in i the spheroid similar to the spheroid 

 A F E D, the centre of which is in c, and the dimensions of 

 which are diminished in the ratio of K c' to K C, the two 

 points i and i may be supposed in the surface of the spheroid. 

 According to Huygens the velocities according to c i and c' i 

 are proportional to these lines j the times employed in passing 

 through them therefore are equal. Thus the time of the 

 transmission of the light in the broken line oci is equal to 

 unity, as in the broken line oci: the differential of these two 

 times therefore is null, which is the principle of Fermat. 

 ^^ T''^if°"'°^ It is clear, that this reasoning is generally applicable, what- 

 any spheroid. ^^^^ ^® *^® nature of the spheroid, and the position of the 

 points c and c' on the face of the crystal j even if they be not 

 in the right line CK, provided they be infinitely near it. 

 The hypo- Reversing the expression of the velocity, thd principle 



gens, though °^ Fermat gives that of the least action. The laws of refrac- 

 false, repre- tion arising from the hypotheses of Huygens, therefore, are 

 sent the fact. 



generally 



