582 



NA TURE 



{April 22, 1886 



In the Nouveaii BiiUdin des Sciences, par la Societe PliUo- 

 mathique de Paris, tome ii., Paris, 1810, occurs this passage :— 



" Mathematiques.— Sur les Equations differentielles des 

 Courbes du Second Degre, par M. IVIonge. L' equation gene- 

 rale des courbes du second degre etant 



Ay"^ + zBxy + Cx"' + iDy ^ Ex + I -o, 

 dans laquelle A, B, C, D, E sont des constantes, M. Monge 

 donne I'equation diflferentielle debarrassee de toutes ces con- 

 stantes, et il parvient a I'equation suivante, du cinquieme ordre, 



(A) 90/-/ - 45'/" -1-40'" = °> 



les quantites r, s, t, etant definies par les equations suivantes : 

 ciy - ^, dp _ ^ dq 



= /. -f = ?. 



d_r _ ^ dj_ _ f 

 dx dx 



" II faut ensnite voir I'usage de I'equation (A), pour trouver 

 I'integrale d'une equation d'un ordre inferieur qui satisfait a 

 cette equation (A) ; ainsi etant donnee I'equation differentielle 

 (i -\- p")y = 3/17-, il parvient a I'integrale (.(- - a]'^ + (y - bf =c-, 

 qui est I'equation d'un cercle. 



"La meme methode pourroit s'appliquer aux equations des 

 lourbes d'un degre superieur au second." 



A note is added to the ,efrect that "Get article est extrait de 

 la Correspondance de I'Ecole imperiale Polytechnique, redigee 

 par M. Hachette : ir cahier du 2e volume, 1810." The pre- s 

 mark of this work at the British Museum is PP. I543' 



Trusting that this is the reference you are in search of, and 

 that the long delay in the discovery of it may be excused when 

 the difficulty of identifying a particular passage (known perhaps 

 only in its full extent to those whose chief work is concerned 

 with such matters) is considered. 



I remain, Sir, faithfully yours, 

 H. Fisher 



Prof. J. J. Sylvester, &c., &c. 



New College, Oxford, .'\pril 19 J. J. Sylvester 



On the Velocity of Light as Determined by Foucault's 

 Revolving Mirror 



It has been shown by Lord Rayleigh and others that the 

 velocity (U) with which a group of waves is propagated in any 

 medium may be calculated by the formula — 



d\ogV \ 



d log \ /' 



where V is the wave-velocity, and \ the wave-length. It has 

 also been observed by Lord Kayleigh that the fronts of the waves 

 reflected by the revolving mirror in Foucault's experiment are 

 inclined one to another, and in consequence must rotate with 

 an angular velocity — 



d\ "' 



U= V 



such experiments depend upon the value of U, and not upon 

 that of V. 



The discussion of the experiment by following a single wave, 

 and taking account of its rotation, is a complicated process, and 

 one in which it is very easy to leave out of account some of the 

 elements of the problem. The principal objection to it, how- 

 ever, is its unreality. If the dispersion is considerable, no wave 

 which leaves the revolving mirror will return to it. The indi- 

 vidual disappears, only the group has permanence. Prof. 

 Schuster, in his communication of March 11 (p. 439), has 

 nevertheless obtained by this method, as the quantity determined 

 by " the experiments hitherto performed," f-/(2K- t'), which, 

 as he observes, is nearly equal to U. He would, I think, have 

 obtained U precisely, if for the angle between two successive 

 wave-planes of similar phase, instead of aze/A/ V, he had used the 

 more exact value 7,ivKJ U. 



By the kindness of Prof. Michelson, I am informed with 

 respect to his recent experiments on the velocity of light in 

 bisulphide of carbon that he would be inclined to place the 

 maximum brilliancy of the light between the spectral lines D 

 and E, but nearer to D. If we take the mean between D and 

 E, we have — 



where a is the angle between two successive wave-planes of similar 

 phase. When dVjdX is positive (the usual case), the direction 

 of rotation is such that the following wave-plane rotates towards 

 the position of the preceding (see Nature, vol. xxv. p. 52). 



But I am not aware that attention has been called to the im- 

 portant fact, that while the individual wave rotates the wave- 

 normal of the group remains unchanged, or, in other words, that 

 if we fix our attention on a point moving with the group, there- 

 fore with the velocity U, the successive wave-planes, as they pass 

 through that point, have all the same orientation. This follows 

 immediately from the two formulae quoted above. For the 

 interval of time between the arrival of two successive wave- 

 planes of similar phase at the moving point is evidently 

 \I(V - U), which reduces by the first formula to dKldf. In 

 this time the second of the wave-planes, having the angular 

 velocity adV/dK, will rotate through an angle a towards the 

 position of the first waveq^lane. But o is the angle between the 

 two planes. The second plane, therefore, in passing the moving 

 point, will have exactly the same orientation which the first had. 

 To get a picture of the phenomenon, we may imagine that we 

 are able to see a few inches of the top of a moving carriage- 

 wheel. The individual spokes rotate, while the group maintains 

 a vertical direction. 



This consideration greatly simplifies the theory of Foucault's 

 experiment, and makes it evident, I ihink, that the results of all 



1745. 



7i(2V - U) ^ J. 



V- 



737- 



A' denoting the velocity //; vacuo (see Am^r. your. Set., vol. 

 xx.xi. p. 64). The number observed was I 76, " with an uncer- 

 tainty of two units in the second place of decimals." This 

 agrees best with the first formula. The same would be true 

 if we used values nearer to the line D. 



J. Willard Gibes 

 New Haven, Connecticut, April i 



The Effect of Change of Temperatureoon the Velocity 

 of Sound in Iron 



I venture to draw attention to an error relating to the above 

 subject, which, originating with Wertheim, still holds a place 

 in some of our modern books on science. According to 

 Wertheim, the velocity of sound in iron and steel is increased 

 by rise of temperature not extending beyond 100° C. Now in 

 n 1 sense whatever is this statement correct. It is true that the 

 longitudinal elasticity of iron, as determined by the static 

 method, will be found greater at loo° C. than at 0° C, pro- 

 vided we begin with i/ie ioiuer temperature first and the wire 

 has not, after the original annealing, been previously raised to 

 100° C. ; but the apparent temporary increase of elasticity is 

 really a permanent one [Phil. Trans., part i., 1883, pp. 129- 

 131), and if the wire be repeatedly heated to 100'' C. and after- 

 wards cooled, subsequent tests will always show a less elasticity 

 at the higher temperature than at the lower, \{ suffici<nt rest after 

 cooling be allowed. When, however, we come to such small 

 molecular displacements as are involved in the passage of sound 

 through a wire, even the apparent increase of elasticity men- 

 tioned above vanishes. I have been able to prove that, when 

 an iron or steel wire is thrown into longitudinal vibrations, so 

 as to produce a musical note, the pitcli of this note becomes 

 lower as we raise the temperature, even when the wire is heated 

 for the first time after it has left the maker's hands. 



It seems rather strange that this er-ror should have so long 

 been allowed to remain uncor-rected, for it has been known for 

 many years that the pitch of a tuning-fork made of steel is 

 lowered by small rises of temperature, and the main part of 

 this lowering must be due to the decrease of elasticity of the 

 steel. Herbert Toiilinson 



King's College, Strand, April 10 



Sound-producing Apparatus of the Cicadas 

 With regard to the above subject, treated of in an article by 

 Mr. Lloyd Morgan in February last (Nature, February 18, p. 

 368), I may mention that some time ago I examined the drum 

 of the comn^ou cicadas found plentifully in the Himalaya near 

 Simla, and which in the evenings send forth a deafening roar 

 from the rhododendron-trees like the whirr of large machinery. 

 Generally the arrangement of the drum and the powerful 

 muscles was as figured by Mr. Morgan, but I also noticed the 

 following particulars not mentioned by him. 



The chitinous rods in the membrane of the drum were not 

 parallel, but converged slightly towarxls one point of the mem- 



