EXPLANATION OF A DYNAMICAL PARADOX,

[From the Messenger of Mathematics, New Series, 1, 1872, pp. 1-3.]

THE answer to the following question, proposed in the Smith's Prize Examination, 1871, is sent in compliance with a request from one of the Editors:

In a compound pendulum consisting of masses m, m' attached to strings of length 1, l', in which of course the most general small motion in one plane consists of two harmonic vibrations superposed, if the upper mass m be very large compared with the under mass m', it is clear that one of the two periodic times (that corresponding to the mode of vibration in which m is nearly at rest) must be very nearly the same as in a simple pendulum of length l', and the other very nearly the same as in a simple pendulum of length l. By a continuous variation of l', the former may be made to pass continuously from less to greater than the latter, and therefore for some value of l' nearly equal to l the two must be equal. But when a system is in stable equilibrium (as is clearly the case here) the equation the roots of which give the times of vibration cannot have equal roots, for that would imply the transitional condition between stable and unstable.

'Point out precisely the fallacy which leads to the above contradiction."

The fallacy lies in the tacit assumption that it is the same root of the quadratic which determines the times of vibration that correspond throughout to the same approximate physical state, ie. the state in which (not considering the special case in which l, l' are nearly equal) the upper mass is nearly at rest, or the two masses move through comparable spaces, as the case may be. Let T, T' be the two times of vibration (or half periods), 7, 7′ the times of vibration of simple pendulums of lengths l, l'; and suppose m, m', l, l' to change continuously, yet so that 7 always remains distinctly greater than or distinctly less than '; i.e. so that the ratio of l~l' to l or l', though absolutely, it may be, small, remains finite while m' m may be taken as small as we

T

please. Of the two T, T', let T be that which for one set of values of the constants m, m', l, l' is nearly equal to 7; then must the same root T remain throughout that which is nearly equal to T; for it is obliged to be nearly equal to one of the two T, T which do not become nearly equal to each other. But if we suppose I, l' to change continuously, so that I, from having been distinctly less, becomes distinctly greater than l', and if T be that root which for l<l' is nearly equal to 7, since 7, 7' pass through equality, and T is merely known to be nearly equal to one of them, there is nothing to shew which of the two T, 7' it is that T is nearly equal to when l>l. By the general principle referred to in the second part of the question, we see that it must be 7'.

The same thing may of course be shewn by the direct solution of the problem. Putting n for π/T, we find by the usual methods mll'na — (m+m') (l + l' ) gn2 + (m + m' ) g2 = 0,

g being gravity, and the roots of this quadratic in n2 are

2

If m' be very small, and l, l' not very nearly equal, the radical becomes very nearly 1/1/'; and denoting by n2, n' the roots corresponding to the signs +, respectively, we have very

When 1, l'are nearly equal, we can no longer distinguish the two harmonic vibrations by the character, that from one of them m is nearly at rest, while the small mass m' moves considerably, since the motion of m as compared with m' is comparable in the two. In fact, the harmonic vibrations, which, when 7, l' are distinctly different, are characterized by the properties above referred to, have their properties interchanged when l:l' passes through 1*.

[* The two roots found above obviously cannot be equal. The limitations to equality of roots being an indication of instability have been determined by Weierstrass and by Routh; cf. Thomson and Tait, Nat. Phil. ed. 11, § 343.]

ON THE LAW OF EXTRAORDINARY REFRACTION IN

ICELAND SPAR.

[From Proceedings of Royal Society, xx, pp. 443-4. Received June 20, 1872.]

It is now some years since I carried out, in the case of Iceland spar, the method of examination of the law of refraction which I described in my report on Double Refraction, published in the Report of the British Association for the year 1862, p. 272*. A prism, approximately right-angled isosceles, was cut in such a direction as to admit of scrutiny, across the two acute angles, in directions of the wave-normal within the crystal comprising respectively inclinations of 90° and 45° to the axis. The directions. of the cut faces were referred by reflection to the cleavage-planes, and thereby to the axis. The light observed was the bright D of a soda-flame.

The result obtained was, that Huygens's construction gives the true law of double refraction within the limits of errors of observation. The error, if any, could hardly exceed a unit in the fourth place of decimals of the index or reciprocal of the wavevelocity, the velocity in air being taken as unity. This result is sufficient absolutely to disprove the law resulting from the theory which makes double refraction depend on a difference of inertia in different directions†.

I intend to present to the Royal Society a detailed account of the observations; but in the mean time the publication of this preliminary notice of the result obtained may possibly be useful to those engaged in the theory of double refraction ‡.

[† Cf. supra, p. 182.]

+

+

[* Supra, p. 187.] [This work was followed up by an investigation by Glazebrook, Phil. Trans. CLXXI, 1880, pp. 421-449, with Prof. Stokes' apparatus, which verified the same degree of accuracy over a wide range of measurements, a limit being imposed by imperfections in the lenses employed, and by some measures by Abria and by Kohlrausch also of the same degree of accuracy. Finally C. S. Hastings, American Journal of Science, cxxxv, Jan. 1888, pp. 60–73, pushed the verification within two units in the sixth place of decimals in two measurements on Iceland spar. In the work of Abria, referred to in the next paper, the uncertainty was about one per cent.]

SUR L'EMPLOI DU PRISME DANS LA VÉRIFICATION DE LA LOI

DE LA DOUBLE RÉFRACTION.

[From Comptes Rendus, Vol. LXXVII, Nov. 17, 1873, pp. 1150–1152.]

LA Communication de M. Abria (Comptes rendus, séance du 13 octobre, p. 814 de ce volume) me détermine à appeler l'attention de l'Académie sur une méthode que j'ai proposée pour le même objet dans un travail sur la double réfraction*, et que j'ai appliquée plus tard au spath calcairet. Cette méthode me paraît plus facile, plus générale et plus exacte que celle de M. Abria.

Quand on veut mesurer l'indice de réfraction d'une substance ordinaire, on emploie le plus souvent la méthode de la déviation minimum. Mais il y a une autre méthode, aussi exacte et presque aussi facile, qui consiste à mesurer la déviation pour un azimut arbitraire du prisme, et en outre l'angle d'incidence ou l'angle d'émergence, suivant que le prisme demeure en repos quand on déplace la lunette, ou qu'il l'accompagne dans son mouvement. Cette méthode n'est pas nouvelle: elle a déjà été employée par M. Swan dans sa vérification de la loi de Snellius pour le rayon ordinaire du spath calcaire‡; mais on n'avait pas, à ma connaissance, indiqué le parti qu'on en pourrait tirer pour la recherche de la loi de la réfraction extraordinaire dans les cristaux. Le phénomène que l'on observe dans le cas d'un cristal est le même que dans le cas d'une substance ordinaire, avec cette seule différence que l'on obtient deux images au lieu d'une seule; on peut encore mesurer la déviation de chacune des deux images, et il ne s'agit que d'interpréter les résultats obtenus. Or, en s'appuyant sur la démonstration qu'a donnée Huyghens pour la

* Report of the British Association for 1862, Part 1, p. 272. [Supra, p. 187.]

+ Proceedings of the Royal Society, Vol. xx, p. 443 (20 juin 1872). [Supra, preceding page.]

‡ Transactions of the Royal Society of Edinburgh, Vol. xvi, p. 375.

S. IV.

22

réfraction en général, démonstration qui, fondée sur le seul principe de la coexistence des petits mouvements, n'exige aucune hypothèse sur la loi de variation des vitesses de propagation dans diverses directions, on démontre facilement que les deux quantités qui représentent pour une substance ordinaire, 1o l'angle de réfraction, 2o l'indice de réfraction, et qui se déduisent des données d'observations par un calcul très-facile, expriment pour un cristal, 1o l'inclinaison de l'onde réfractée à la surface d'incidence, onde qui est nécessairement perpendiculaire au plan d'incidence, 2o le rapport de la vitesse de propagation dans l'air à celle de l'onde réfractée. La direction ainsi déterminée par rapport aux deux faces du prisme est rapportée ensuite, par le calcul, à des directions fixes dans le cristal, l'orientation de chaque face artificielle ayant été déterminée au moyen de la réflexion, par rapport à des faces, soit naturelles, soit de clivage. On peut ainsi examiner un cristal dans une série de directions, au moyen d'un seul angle réfringent, et l'on peut faire tailler deux angles au moins sur un même bloc sans détruire les faces dont on a besoin pour la détermination de l'orientation des plans artificiels.

Je n'ai appliqué jusqu'ici cette méthode qu'au spath calcaire, cristal que j'ai choisi à cause de la facilité avec laquelle on peut s'en procurer de bons échantillons, et de l'énergie de sa double réfraction, qui devrait rendre plus sensibles les écarts par rapport à la loi d'Huyghens, s'il en existait. J'ai trouvé que cette loi représente la réfraction extraordinaire aussi exactement que la loi de Snellius représente la réfraction ordinaire.

L'erreur moyenne de quinze observations du rayon extraordinaire, faites dans des directions qui s'étendaient de 30 à 60 degrés environ de l'axe, et rapprochées de la formule déduite de la construction d'Huyghens en y introduisant les indices principaux, obtenus à 90 degrés de l'axe, ne s'élevait qu'à 0,00013 de l'indice, quantité qui est de l'ordre des erreurs accidentelles de mes observations, et qui correspond à 13 environ de la différence des indices principaux. L'erreur correspondante de déviation dans un prisme de 45 degrés est d'environ 25 secondes.

1 1300