206 Notes on some Points in the Theory of Light. 



tion still leaves the demonstration perfectly rigorous in the case 

 of circular vibrations, and does not affect its force when the 

 vibrations are elliptical. For in the rotatory fluids it is obvious 

 that the normal vibrations, supposing such to exist, must, by 

 reason of the symmetry which the fluid constitution requires, be 

 independent of the transversal vibrations, and separable from 

 them, so that the one kind of vibrations may be supposed to 

 vanish when we wish merely to determine the laws of the other. 

 The equations (2) are, therefore, quite exact in this case; and 

 they are also exact in the case of a ray passing along the axis 

 of quartz, since such a ray is not experimentally distinguishable 

 from one transmitted by a rotatory fluid, and its vibrations must 

 consequently be subject to the same kind of symmetry. In these 

 two cases, therefore, it is rigorously proved that the values of A*, 

 which ought to be equal to plus and minus unity, are imaginary, 

 and equal to f - 1. And if we now take the most general 

 case with regard to quartz, and suppose that the ray, which was 

 at first coincident with the axis of the crystal, becomes gradually 

 inclined to it, the values of k must evidently continue to be ima- 

 ginary, until such an inclination has been attained that the two 

 roots of equation (5) become possible and equal, in consequence 

 of the increased magnitude of the co-efficient of the second term. 

 Supposing the last term of that equation to remain unchanged, 

 this would take place when the co- efficient of k (without regard- 

 ing its sign) became equal to the number 2, and the values of k 

 each equal to unity, both values being positive or both negative. 

 The vibrations which before were impossible would, at this in- 

 clination, suddenly become possible ; they would be circular, 

 which is the exclusive property of vibrations transmitted along 

 the axis ; and they would have the same direction in both rays, 

 which is not a property of any vibrations that are known to 

 exist. At greater inclinations the vibrations would be ellip- 

 tical, but they would still have the same direction in the two 

 rays. These results would not be sensibly altered by regard- 

 ing the equation (5) as only approximate in the case of rays 

 inclined to the axis ; for the last term of that equation, if it 



