Notes on some Points in the Theory of Light. 205 



turpentine ; the position of the co-ordinates x and y, in the 

 plane of the wave, being now, of course, arbitrary. In each 

 of these cases we have k = 1, and A = B = a 2 , so that the 

 value of s 2 in equation (6) is expressed by the constant a 2 , 

 plus or minus a term which is inversely proportional to the 

 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 2 from the formulas (4), unless 

 we suppose A' = B' = a?, since it is only in the expansion of 

 (7 that a term inversely proportional to A can be found ; but 

 on this supposition the formulas are inconsistent with each 

 other, nor can they be reconciled by any value of k. Indeed, 

 when A' = B', the equation (5) give k = */ - 1. Thus it 

 appears that circular vibrations, such as are known to be propa- 

 gated along the axis of quartz, and through certain fluids, can- 

 not 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 ex- 

 actly circular, but in some degree elliptical a supposition which, 

 if it were at all conceivable, which we have seen it is not (p. 196), 

 would be at once set aside by what has just been proved; for no 

 assumed value of #, whether small or great, will in any way help 

 to remove the difficulty. 



But this is not all. Eectilinear vibrations are excluded as 

 well as circular ; for we cannot suppose k = in the equations 

 (4), so long as the quantity C', resulting from the hypothesis of 

 unsymmetrical arrangement, has any existence. Thus the in- 

 consistency of that hypothesis is complete, and the equations to 

 which it leads are utterly devoid of meaning. 



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 intelligible, and to get rid of 

 M. Cauchy's " third ray," which has no existence in the nature 

 of things, I have suppressed the normal vibrations ; a procedure 

 which is not, in general, allowable on the principles of M. 

 Cauchy. It will readily appear, however, that this simplifica- 



