2 54 



NATURE 



[July 12, 1883 



molecules is on the crest of the wave, or has reached its 

 maximum displacement ; that one (Fig. 5) showing the mag- 

 nitude and direction of the velocities, after the wave has 

 advanced such a distance, as (in this case equal to one- 

 twenty-fourth 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 



Fig. 9. 

 Two particles in Wave-Length. 



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' hitherto 

 unpublished explanation of phosphorescence. 



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 mike 

 the influence of coarse-grainedness on the velocity of 

 propagation smaller than it is on the simple assumption, 

 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 calcu- 

 lated results, you will see that with as few as 20 molecules in 

 the wave-length, the velocity of propagation is 99! per cent, 

 of what it would be with an infinite number of molecules ; 

 hence the extreme difference of propagational velocity, 

 accountable for by Cau:hy's unmodified theory in its 

 idealised extreme of mutual action limited to nearest 

 neighbours, amounts to 1/200. 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 1/17 ; in dense flint 

 glass to nearly 1/30 ; in hard crown glass to 1/73; and in 

 water and alcohol to rather more than 1/100. Hence, 

 none of these substances can have so many as 20 mole- 

 cules 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 



Table III. — Table of Refractive Indices. 



The numbers in the first two columns were determined by Dr. Hopkinson, 

 those in the last three by Messrs. Gladstone and Dale. The index of 

 refraction of air for light near the line E is 1-000294. 



see that there could not be more than twelve molecules in 

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

 in hard crown glass; eight in flint glass ; and in carbon 

 disulphide actually not more than four molecules in the 

 wave-length, if we are to depend upon Cauchy's unmodi- 

 fied theory for the explanation of dispersion. So large 

 coarse-grainedness of ordinary transparent bodies, solid 

 or fluid, is quite untenable. Before I conclude I intend 

 to show you, from the kinetic theory of gases, a superior 

 limit to the size of molecules, according to which, in 

 glass or in water, there is probably something like 600 

 molecules to the wave-length ; and almost certainly not 

 fewer than two, or three, or four hundred. But even 

 without any such definite estimate of a superior limit 

 to the size of mole:ules, there are many reasons against 

 the admission that it is probable or possible, there 

 can be only four, or five, or six to the wave-length. 

 The very drawing by Nobert of 4,000 lines on a breadth 

 of a millimetre, or at the rate of 40,000 to the centimetre, 

 or about two to the ether wave-length of blue ( F) light, 1 

 seems quite to negative the idea of any such possibility, 

 of only five or six molecules to the wave-length, even if 

 we were not to declare against it from theory and ob- 

 servation of the reflection of light from polished surfaces. 



{To be continued.) 



1 Loschmtdt, "quoting from the Zollvereins department of the London 



International Exhibition of 1862. page 83. and from Harting 'On the 



Microscope.' page 88i," Sitmiigiierickte iter Wiener AkademU Math. 

 Phys. 1865. vol. lii. 



