Crystalline Reflexion and Refraction. 175 



But the second of the equations (31) has another signifi- 

 cation. For if the transversals applied at the extremities of 

 the refracted rays be projected on the plane of # s< which is 

 the plane of incidence, and contains the axes of z z and s' 2 , the 

 projections will be perpendicular to these axes, since the trans- 

 versals themselves are perpendicular to them ; and the distances 

 of the projections from the point will be proportional to 8 

 and s', or, by the relations (29), to sin i 2 and sin / 2 ; so that 

 if 2 and 0' 2 be the angles which the transversals make with 

 the plane of incidence, the moments of the projections will 

 be represented by r 2 cos 3 sin 4 and / 2 cos 0' 2 sin / 2 . At the 

 same time, if 0i and 9\ be the angles which the incident and re- 

 flected transversals make with the plane of incidence, the mo- 

 ments of the corresponding projections of these transversals will 

 evidently be represented by TI cos 0i sin i l9 and - T\ cos 0\ sin ii ; 

 the latter quantity being taken with a negative sign, because 

 the extremity of the reflected ray, where the transversal T\ 

 is applied, lies in the first medium, while the extremities of 

 the incident and refracted rays lie in the second, and it is 

 supposed that when any of the angles O l9 0\, 2 , 0' 2 is zero, 

 the direction of the corresponding transversal makes an acute 

 angle with the axis of o? . Hence we have 



TI cos 0i sin h - T\ cos 9\ sin ^ = r 2 cos 2 sin i z + r' 2 cos 0' 2 sin i\ ; 



an equation which expresses that if each transversal be pro- 

 jected upon the axis of s 0) the sum of the projections of the 

 incident and reflected transversals will be equal to the sum 

 of the projections of the refracted transversals. Therefore, 

 since the phases of the different vibrations are identical when 

 Z Q = 0, the condition (22) is fulfilled, as it ought to be. 



On account of this identity of phases, it follows from the 

 conditions (20) and (22), that if the transversals be drawn 

 through the point 0, and those which belong to each medium 

 be compounded like forces acting at a point, their resultants 

 will be the same ; that is, the resultant of the incident and 



