202 Sir William Thomson [Feb. 2, 



position of tlie arrows) the magnitude and direction of velocity of 

 each molecule at the instant when one of the molecules is on the crest 

 of the wave, or has reached its maximum displacement ; that one 

 (Fig. 5) showing the magnitude and direction of the velocities after 

 the wave has advanced such a distance as (in this case equal to l-24th 

 of the wave-length) to bring the crest of the wave to midway between 

 two molecules. This pair of diagrams (Figs. 6 and 7) shows the 

 same for waves having four molecules in the wave-length, and this 

 pair (Figs. 8 and 9) for a wave having two molecules in the wave- 

 length. 



The more nearly this critical case is approached, that is to say, 

 the shorter the wave-length down to the limit of twice the distance 

 from molecule to molecule, the less becomes the difference between 

 the two configurations of motion constituted by waves travelling in 

 opposite directions. In the extreme or critical case the difference is 

 annulled, and the motion is not a wave motion, but a case of what is 

 often called " standing vibration." Before I conclude this evening I 

 hope to explain in detail the kind of motion which we find instead of 

 wave-motion (become mathematically imaginary), when the vibrational 

 period of the exciter is anything less than the critical value, because 

 this case is of extreme importance and interest in physical optics, 

 according to Stokes's hitherto unpublished explanation of phospho- 

 rescence. 



This supposition of each molecule acting with direct force only 

 on its nearest neighbour is not exactly the postulate on which Cauchy 

 works. He supposes each molecule to act on all around it, according 

 to some law of rapid decrease as the distance increases ; but this must 

 make the influence of coarse-grainedness on the velocity of propaga- 

 tion smaller than it is on the simple assumj)tion realised in the 

 models and diagrams before you, which therefore represents the 

 extreme limit of the efficacy of Cauchy's unmodified theory to 

 explain dispersion. 



Now, by looking at the little table (Table II.) of calculated 

 results, you will see that, with as few as 20 molecules in the wave- 

 length, the velocity of propagation is 99 J per cent, of what it would 

 be with an infinite number of molecules ; hence the extreme difference 

 of propagational velocity, accountable for by Cauchy's unmodified 

 theory in its idealised extreme of mutual action limited to nearest 

 neighbours, amounts to l-200th. Now look at this table (Table III.) 

 of refractive indices, and you see that the difference of velocity of 

 red light A, and of violet light H, amounts in carbon disulphide to 

 1-1 7th ; in dense flint glass to nearly l-30th; in hard crown glass 

 to l-73rd; and in water and alcohol to rather more than 1-lOOth. 

 Hence, none of these substances can have so many as 20 molecules in 

 the wave-length, if dispersion is to be accounted for by Cauchy's 

 unmodified theory, and by looking back to the little table of calculated 

 results (Table II.), you will see that there could not be more than 

 12 molecules in the wave-length of violet light in water or alcohol ; 



