erystalline Reflexion and Refraction. 43 
an equation which expresses that if each transversal be projected upon the axis of 
%» 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, = 0, the condi- 
tion (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 O, and those which 
belong to each medium be compounded like forces acting at a point, their result- 
ants will be the same ; that is, the resultant of the incident and reflected transversals 
will be the same as the resultant of the refracted transversals. 
Hence, recollecting what has been proved respecting the moments of the 
transversals applied at the extremities of the rays, we have the following theorem : 
Supposing the length of each ray, measured from the point of incidence and 
in the direction of propagation, to be taken proportional to the velocity with 
which the light is propagated along it, andits transversal to be drawn through the 
extremity of this length, the incident and reflected transversals having their proper 
directions, but the refracted transversals having their directions reversed ; if all 
the transversals so drawn be compounded like forces applied to a rigid body, 
their resultant will be a couple, lying in a plane parallel to the plane which sepa- 
rates the two media. 
This theorem affords a complete solution of the question of reflexion and 
refraction.* Expressed analytically it gives five equations, of which four are 
independent. 
To apply the preceding results to a simple case, suppose the second medium, 
as well as the first, to be an ordinary one. We have then only one refracted ray, 
and one refracted transversal 7,. 
1°. When the incident ray is polarized in the plane of incidence, the trans- 
versals are all in that plane; and as they are perpendicular to the rays, and the 
* The same theorem applies to the other case of reflexion and refraction, when a ray which has 
entered the crystal emerges from it into an ordinary medium, undergoing double reflexion at the 
surface where it emerges. In fact, the conditions (20) and (21) hold good whether the ordinary 
medium is the first or the second ; and in the latter case, as well as in the former, it may be shown 
that the condition (22) is fulfilled, and that the theorem above-mentioned is true. 
G2 
