Crystalline Reflexion and Refraction. 183 



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 co-ordinates ; and the right-hand members, as 

 appears by the formulae (35) and (41), are proportional to the 

 quantities 



COS a^ COS P COS y 



- r cos cr 2 , - r cos p 2 , - r cos 72, 



s s s 



which are obviously the differences between the corresponding 

 co-ordinates of the points R and Q. The plane of the transver- 

 sals 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 uniradial ; and as each re- 

 fracted 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 accordance with the laws estab- 

 lished 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 



* Transactions, Royal Irish Academy, VOL. jmii. p. 39 (supra, p. 96). 



