40 Mr. Mac Cullagh on the Laws of 



ing transversal of the reflected ray ? The answer is simple — both transversals 

 will lie in the polar plane of the refracted ray. Let us pursue this remark a 

 little. 



The refracted ray OT being given, we can find its polar plane, and thence 

 the intersections of this plane with the incident and reflected wave planes. 

 These intersections will be the positions of the incident and reflected transversals 

 when OT is the sole refracted ray. The refracted transversal lies also in the 

 polar plane ; and this transversal Is, by our fourth hypothesis, the diagonal of a 

 parallelogram, whose sides are the other two transversals ; which determines the 

 relative lengths of the three transversals, or the relative amplitudes of the 

 vibrations. The intensities of the reflected and incident rays are, of course, 

 proportional to the squares of their transversals. When the ray OT disappears, 

 we must take the polar plane of the other ray, and proceed as before. 



Thus there are, in the incident wave plane, two transversal directions which 

 give only a single refracted ray. These, as wellvas the corresponding ones in 

 the reflected wave plane, may be called uniradial transversals. They are the 

 intersections of the two refracted polar planes with the incident and reflected 

 wave planes. 



When the incident transversal does not coincide with either of the uniradial 

 directions, it is to be resolved parallel to them, and then each component 

 transversal will supply a refracted ray, according* to the foregoing rules. The 

 reflected transversals, arising from the component incident ones, are to be found 

 separately by the same rules, and then to be compounded. 



In ordinary reflexion, if the incident transversal be in the plane of incidence, 

 or perpendicular to it, the reflected transversal will be so likewise. But this 

 does not hold in crystalline reflexion. The general method just given will, 

 however, enable us to determine the positions and magnitudes of the reflected 

 transversals in these two remarkable cases ; and then, if we choose, we can reduce 

 any other case to these two, by resolving the incident transversal in directions 

 parallel and perpendicular to the plane of incidence. 



If we conceive a pair of incident transversals, at right angles to each other, 



to revolve about the origin, it is evident that thei-e will be a position in which 



the reflected transversals corresponding to them will also be at right angles to 



. each other. There is no difficulty in finding this position, and there will be an 



