Crystalline Reflexion and Refraction. 173 



different waves at the time t. To see how they depend on the 

 co-ordinates # , y , z , conceive the axis of z to be directed from 

 towards the interior of the second medium, and the axis of x 

 to lie in the plane of incidence, so that the positive directions of 

 Zj, z 2 , s'j may lie within the angle made by the positive direc- 

 tions of x and z , while the positive direction of z'i lies within 

 the angle made by the positive direction of x and the negative 

 direction of z . Let ij. be the angle of incidence, and i z , i\ the 

 angles of refraction ; then 



Zi = X Q sin i + s cos !, z i = # sm h - z cos !, 



(28) 

 z 2 = x sin i-i + z cos ' a , z' 2 = x sin i\ + z cos i\. 



These values are to be written in the expressions (25). They 

 show that the phases, and therefore the displacements, are in- 

 dependent of i/ . 



Since the conditions relative to the plane of x tyo must hold 

 at every instant of time, and for every point of that plane, the 

 co-efficients of t, as well as those of x , in the values of the 

 different phases, must be identical; so that we must have 



sn e _ sn ? 2 _ sn i 

 AI A 2 ' A 2 



Therefore, when z = 0, the supposition 



Vi = l/t = V 2 = V* (30) 



renders the phases identical, independently of t and a? . And, 

 from the form of the equations of condition, it is easy to see 

 that this supposition is necessary; because the equations (20), 

 when the values (26) are substituted in them, contain only the 

 cosines of the phases ; and the equations (21), when the values 

 (27) are substituted in them, contain only the sines of the 



