3 8 Geometrical Propositions applied 



and so on. In general, every ray going into the crystal from 

 the first surface, whether after refraction or after any even 

 number of internal reflections, is parallel either to OP f or to OM '; 

 and every ray returning from the second surface of the crystal 

 after any odd number of internal reflections, is parallel either to 

 Op' or to Om '. Thus the direction of every ray in the interior 

 of the crystal is the same as the direction of some one of the four 

 lines OP', OM ', Op', Om' ; and the velocity of the ray is in- 

 versely as the length of this line ; so that the velocity of the ray 

 OM', for example, or of any ray parallel to OM ', is to the velo- 

 city V as OS is to OM'. The little plane waves that, keeping 

 always parallel to themselves, move along these rays, are respec- 

 tively perpendicular to the lines OP, OM, Op, Om\ and the 

 lengths of these lines are inversely as the velocities of the waves 

 estimated in directions perpendicular to their planes ; so that the 

 velocity of the wave which moves along the ray OM', or along 

 any parallel ray, is to the velocity V as OS is to OM. 



38. The ray OP', and all the rays parallel to it, are perpen- 

 dicular to the plane which touches at P the surface of refraction ; 

 and the waves which move along these rays are perpendicular to 

 the right line OP. Any ray of this set may be called a ray P, 

 and any of the waves a wave P. In like manner, the rays M,p, m, 

 are rays that are perpendicular to the tangent planes at the 

 points M, p, m, respectively ; and the waves M, p, m, are the 

 waves that belong to these rays, and that have their planes re- 

 spectively perpendicular to the right lines OM, Op, Om.- The 

 rays P, M, all come from the first surface of the crystal ; the 

 rays p, m, from the second. 



As the ordinates RP, Rp, are greater than the ordinates 

 RM, Rm, so the rays P, p, are more refracted or more reflected 

 than the rays M, m. The former rays may therefore be said to 

 be plus refracted, or plus reflected, and the latter to be minus re- 

 fracted, or minus reflected. Or for the convenience of naming 

 the rays P, p, may be called plus rays ; and the rays M, m, 

 minus rays. The waves P, p, in like manner, may be termed 

 phis waves, and the waves M, m, minus waves. 



