151 



wave-length A; the sign of this term depending on the direc- 

 tion of the circular vibration. Now it will not be possible 

 to obtain a similar value of s^ from the formulas (4), unless 

 we suppose a' = b' = a^, since it is only in the expansion 

 of c' that a term inversely proportional to X can be found ; 

 but on this supposition the formulas are inconsistent with 

 each other, nor can they be reconciled by any value oi /c. 

 Indeed, when a' = b', the equation (5) gives k — ± \/ — [. 

 Thus it appears that circular vibrations, such as are known 

 to be propagated along the axis of quartz, and through cer- 

 tain fluids, cannot possibly exist on the hypothesis of M. 

 Cauchy. It was probably some partial perception of this 

 fact that caused M. Cauchy to assert that the vibrations, 

 in these cases, are not exactly circular, but in some degree 

 elliptical; a supposition which, if it were at all conceivable, 

 which we have seen it is not (p. 142), would be at once set 

 aside by what has just been proved; for no assumed value 

 oUc, whether small or great, will in any way help to remove 

 the difficulty. 



But this is not all. Rectilinear vibrations are excluded 

 as well as circular; for we cannot suppose ^ z= in the equa- 

 tions (4), so long as the quantity c', resulting from the hypo- 

 thesis of unsymmetrical arrangement, has any existence. 

 Thus the inconsistency of that hypothesis is complete, and the 

 equations to which it leads are utterly devoid of mean- 

 ing. 



The foregoing investigation does not differ materially 

 from that which I had recourse to in the beginning of the 

 year 1836. To render the proof more easily intelHgible, 

 and to get rid of M. Cauchy's "third ray," which has no 

 existence n the nature of things, I have suppressed the 

 normal vibrations ; a procedure which is not, in general, al- 

 lowable on the principles of M. Cauchy. It will readily ap- 

 pear, however, that this simpHfication still leaves the demon- 

 stration perfectly rigorous irt the case of circular vibrations. 



