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and does not affect its force when the vibrations are elHpti- 

 cal. 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 indepen- 

 dent 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 dis- 

 tinguishable 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 k, which ought to be equal to plus 

 and minus unity, are imaginary, and equal to =t V — 1. 

 And if we now take the most general case with regard to 

 quartz, and suppose that the ray, which was at first coinci- 

 dent with the axis of the crystal, becomes gradually inclined 

 to it, the values of ^ must evidently continue to be imaginary, 

 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 ^ 

 (without regarding 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 inclination, suddenly become pos- 

 sible ; they would be circular, which is the exclusive pro- 

 perty 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 elliptical, but they would 

 still have the same direction in the two rays. These re- 

 sults would not be sensibly altered by regarding the equa- 



