crystalline Reflexion and Refraction. 49 
and from the product of the first and fourth, combined with the product of the 
second and third, 
27,7; sin (6; — @,)cos?,= 7,” tan?,cos 0, { — sin 6, sin 7, + s*(sin 6, sini, —cosi,tane)}. 
In the three equations just found, the left-hand members are proportional, as 
we have seen, to the cosines of the angles which a right line perpendicular to the 
plane of the transversals makes with the axes of coordinates ; and the right-hand 
members, as appears by the formula (35) and (41), are proportional to the quan- 
tities 
COs a cos 6, 
cos 
“ — 1 COS dy, oe 
—T COs B,, — 1 COS ¥.5 
s 
which are obviously the differences between the corresponding coordinates of the 
points R and Q. The plane of the transversals is therefore perpendicular to the 
right line QR, which joins those points. 
A plane parallel to the right line TM, and passing through the transversal 
of the ray OT, is that which I have called the polar plane of the ray ;* and this 
plane is perpendicular to QR. Therefore, when there is only one refracted ray, 
the incident and reflected transversals lie in the polar plane of that ray ; and 
their directions being thus determined, the relative magnitudes of the three 
transversals are known. In this case the incident and reflected transversals are 
called wniradial ; and as each refracted ray in turn may be made to disappear, 
there are two uniradial directions in the plane of the incident wave, and two in 
that of the reflected wave. 
When the incident transversal is not uniradial, it may be considered as the 
resultant of two uniradial transversals, each of which will supply a refracted ray, 
and will produce a uniradial component of the reflected transversal. 
It is needless to extend these deductions further. They have been carried 
far enough to show that the results of the foregoing theory are in perfect accord- 
ance with the laws established in my former paper on the subject of crystalline 
reflexion. The theory itself suggests much matter for consideration ; but at 
present we shall confine ourselves to one remark, which may be necessary to 
* Ibid, p. 39. 
VOL. XXI. H 
