Crystalline Reflexion and Refraction. 173 



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

 co-ordinates x w y m 2 , conceive the axis of z to be directed from 

 towards the interior of the second medium, and the axis of X Q 

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

 21, 2 2 > s'a may lie within the angle made by the positive direc- 

 tions of X Q and 2 , while the positive direction of z\ lies within 

 the angle made by the positive direction of X Q and the negative 

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

 angles of refraction ; then 



21 = x<> sin n + s cos ii t z\ = # sin h - Z Q cos i it 



(28) 

 z z = X Q sin iz + s cos i z , z 2 = X Q sin i 2 + 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 y Q . 



Since the conditions relative to the plane of # ?/o must hold 

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

 co-efficients of t, as well as those of a? > in the values of the 

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



1 _ s s' sin ii _ sin i t _ sin i' z 



AI A2 A 2 AI \z A 2 



Therefore, when Z Q = 0, the supposition 



Vi = l/i = V Z = l/ 2 (30) 



renders the phases identical, independently of t and X Q . 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 



