266 M. Cauchy's Vie^jo of the Undulatory Theory of Light. 

 Now, to show that the velocity varies for waves of different 



o 



lengths is the same thing as to show that the ratio -v- is dif- 

 ferent for different values of k, that is, of /; and this we can 

 determine from the expression in the form in which we now 

 have it. If, for instance, from the nature of the quantities, 

 the last factor should not vary with a change in /, then the 

 requisite condition will not be fulfilled. Now, the variable 

 factor expresses the ratio of a sine to its arc, and this will be 

 very nearly constant for a variation in / if the arc be ex- 

 tremely small, that is, if —t- be very small ; or, in other words, 



tf the ratio of (r) the distance between two molecules to (/) 

 the length of a wave be very small. 



It follows, then, with regard to the hypothetical nature of 

 the aethereal medium, that if the interval between two mole- 

 cules be very much less than the length of a wave, then the 

 velocity will not sensibly vary with the length of a wave. 



In adopting the theory, then, with a view to its application 

 to the facts, we must carefully observe the limitation thus 

 imposed upon the primary nature of our hypothesis. It is a 

 limitation which is perfectly admissible as regards any of the 

 preceding deductions; and we must introduce it as an express 

 condition, that a relation between the velocity and the length of 

 a wave is established on M. Cauchy^s principles, provided the 

 molecules are so disposed that the intervals between them al- 

 ways bear a sensible ratio to the length of an undulation. 



The necessity of the fulfilment of this condition was the 

 suggestion alluded to at first, and on the nature and import- 

 ance of it, I made a few remarks in the Physical Section of the 

 British Association at the Edinburgh meeting. But the ex- 

 pression when reduced to this form presents for examination 

 other points of still higher interest. 



The existence in general of a relation between the length 

 of a wave and the velocity of its propagation (as already ob- 

 served,) assigns a reason why rays whose waves are of dif- 

 ferent lengths should be unequally refracted. But it becomes 

 important, with a view to the more exact comparison of 

 theory and observation, to be able to assign a more specific 

 relation. Indeed the theory will be incomplete unless it enable 

 us to show not only that some relation subsists between the 

 length and the velocity of a wave, which shall vary both for 

 each different ray and each different medium, but also that it 

 is such as shall explain why the several rays are unequally 



