UNDULATION — DISPERSION. 155 



in a state of dilatation, is the same as Avlien it is compressetT. But compression 

 develo])e3 heat, and the expansive force increases directly as the temperature. 

 On this account it is necessary to introduce into the expression for tlie value of 

 V another coiistant, which is the quotient which arises from dividing the capacity 

 of air for heat when it is expanded under a constant pressure, by its capacity 

 when its pressure is raised under a constant volume. These capacities being 

 represented by c' and c, the velocity becomes 



in Avhich all the factors are still constant, and the velocity is uniform. 



It will be further observed that the demonstration is independent- of the am- 

 plitude of the vibration, and also of the length of the wave. The distance AA' 

 has no necessary relation to either the one or the other of those dimensions. 

 The waves will be longer as the ti/ne occupied in a vibration is greater, and 

 shorter as the time is less ; but this will be onl}^ because, in the first case, a 

 larger number of tremors by condensation occur, before the tremors by rarefac- 

 tion commence, than in the second. These tremors, in longer and sliorter waves, 

 are arranged in larger or smaller groups, but every tremor, of whichever species, 

 advances with the same velocity. 



Observation proves that in sonorous waves this theoretic iulluencc is true. 

 In the range of musical tones, the waves corresponding to the deepest notes are 

 two or three hundred times longer than those belonging to the highest ; yet at 

 any distance at which music is audible all the notes of a melody come to the 

 car without the slightest perceptible disturbance of their order. 



As we shall presently have to apply the undulatory theory to the explana- 

 tion of optical phenomena, this seems to be the proper place to anticipate au 

 objection which has been made to such an application of it, founded upon this 

 presumed constancy of velocity of propagation for waves of all lengths. The 

 dispersion of light by refraction, or the separation of the elementary components 

 of light by the prism, must, upon any theory, be regarded as irrefutable evi- 

 dence that the velocities of these components are unequal.* If the undulatory 

 theory be the true one, it is demonstrable that the undulations of the more re- 

 frangible rays are shorter than those of the less refrangible; and it is also neces- 

 sary, on that theory, to admit that in point of fact the A-eiocitics of the same 

 rays are less. This fact so directly conflicts with the proposition just now 

 demonstrated, that for a long time it was regarded, and by some continues still 

 to be regarded, as au almost insuperable objection to the wave theory of light. 

 If we refer once more, however, to the demonstration, we shall see that it 

 involves one assumption which is not strictly true — the assumption tli.'it AA' is 

 too insignificant a quantity to be regarded when subtracted from AB, and in 

 therefore making AB=A'B. In the case of acoustic waves this assumption 

 is admissible, and observation proves that it introduces no sensible error; but 

 if the wave theory of light be the true one, the undulations themselves must be 

 excessively minute; so that it is not only quite possible, but even probable, that 

 the amplitude of the molecular excursions may have a very sensible ratio to the 

 undulation length. Observe, however, that the admission of this supposition 

 does not draw after it the consequence that the velocity of wave propagation 

 must, as a necessity, vary with the varying lengths of the waves; it only viti- 

 ates our previous demonsti'ation that this velocity is constant for waves of all 

 lengths. The same consiaucy may continue to exist, but we must establish the 

 truth, if it is one, by diflercnt methods of proof. It is even possible that it may 

 exist under certain circumstances — that is, in certain media, and not in others. 

 This last is the conclusion which was reached by Mr. Cauchy after an examina- 



* Tliis must at least be true after refraction if uot befure. 



