June 14, 1888] 



NATURE 



165 



these physicists and of Charpentier. — Meteorological observations 

 made at the Brera Observatory, Milan, daring the month of 

 April. 



Rivista Scientifico-Industriale, May 15. — Remarks on the 

 earthquake at Florence on November 14, 1887, by Prof. P. G. 

 Giovannozzi. Following the system adopted by Serpicri, the 

 author has collected data from various quarters showing that the 

 disturbance was of a purely local character. The chief shock, 

 although s i violent as to have been heard by the deaf, passed 

 through the city with such velocity that very little damage was 

 done. It presented all the characters of a true gaseous explosion, 

 taking a vertical direction from a moderate depth below the 

 crust of the earth, and absolutely unconnected with any volcanic 

 phenomena. It is noteworthy that the earthquake followed a 

 long and exceptional period of wet weather, during which a 

 rainfall of 225mm. was recorded within the zone of disturbance. 



SOCIETIES AND ACADEMIES. 

 London. 



Royal Society, May 17.— "On /Folotropic Elastic Solids." 

 By C. Chree, M.A., Fellow of King's College, Cambridge. 

 Communicated by Prof. J. J. Thomson, F.R.S. 



On the multi-constant theory of elasticity, the equations con- 

 necting the strains and stresses contain 21 constants. As shown 

 by Saint-Venant, these reduce for one-plane symmetry to 13, for 

 three-plane symmetry to 9, and for symmetry round an axis 

 perpendicular to a plane of symmetry to 5. 



Part I. of this paper deals with one-plane symmetry. A solution 

 is obtained of the internal equations of equilibrium complete so 

 far as it goes. It is employed in solving the problem, already 

 treated by Saint-Venant, of a beam, whose length is perpen- 

 dicular to the plane of symmetry, held at one end, and at the 

 other acted on by a system of forces, whose resultant consists 

 of a single force along the axis of the beam, and of a couple 

 about any line in the terminal section through its centroid. The 

 case when the cross-section is elliptical, and the beam exposed 

 to equilibrating torsional couples over its ends is also treated. 

 Results are obtained confirmatory of Saint-Venant's. They are 

 also extended to the case of a composite cylinder, formed of 

 shells of different materials whose cross-sections are bounded by 

 concentric similar and similarly situated ellipses, the law of 

 variation being the same for all the elastic constants of the solu- 

 tion. The limiting case of a continuously varying structure is 

 deduced. 



When a beam of circular section is exposed to torsion, it is 

 proved that warping will ensue proportional to the moment of 

 the twisting couple. Only two diameters in the cross-section, 

 and these mutually at right angles, remain perpendicular to the 

 axis of the beam. 



Part II. treats of a material symmetrical round an axis, that 

 of z, and having the perpendicular plane one of symmetry. A 

 general solution of the internal equations of equilibrium is 

 obtained, supposing no bodily forces to act. The solution 

 involves arbitrary constants, and consists of a series of parts, 

 each composed of a series of terms involving homogeneous pro- 

 ducts of the variables, such as x L y m z H ~ l ~ m , where /, m, n are 

 integers, and n is greater than 3. The terms involving powers 

 of the variables, the sum of whose indices is less than 4, are then 

 obtained by a more elementary process, and these alone are 

 required in the applications which follow. 



The first application of the solution is to "Saint-Venant's 

 problem " for a beam of elliptical cross-section. The problem 

 is worked out without introducing any assumptions, and a solu- 

 tion obtained, which is thus directly proved to be the only 

 solution possible if powers of the variables above the third be 

 neglected. 



Part III. consists of an application of the second portion of 

 the solution of Part II. to the case of a spheroid, oblate or pro- 

 late, and of any eccentricity, rotating with uniform angular 

 velocity round its axis of symmetry, which is also the axis of 

 symmetry of the material. The surface of the spheroid is 

 supposed free of all forces. 



The limiting form of the solution, when the polar axis of the 

 spheroid is supposed to diminish indefinitely, is applied to the 

 case of a thin circular disk rotating freely about a perpendicular 

 to its plane through its centre. The solution so obtained is 



shown to satisfy all the conditions required for the circular disk, 

 except that it brings in small tangential surface stresses. 

 According to this solution the disk increases in radius, and 

 diminishes everywhere in thickness, especially near the axis, 

 so as to become biconcave. All, originally plane, sections 

 parallel to the faces become very approximately paraboloids of 

 revolution. 



Again, by supposing the ratio of the polar to the equatorial 

 diameter of the spheroid to become very great, a surface is 

 obtained which differs very little from that of a right circular 

 cylinder. The corresponding form of the solution obtained for 

 the spheroid, when the ratio of the polar to: the equatorial dia- 

 meter becomes infinite, may thus be expected to apply very 

 approximately to a long thin cylinder. This is verified directly, 

 and it is shown that this solution is in all respects as approxi- 

 mately true as that universally accepted for Saiut-Venant's 

 problem. According to the solution the cylinder shortens, and 

 every cross-section increases in radius but remains plane. 



Part IV. treats of the longitudinal vibrations of a bar of uni- 

 form circular section and of material the same as in Part II. 

 Assuming strains of the form — 



radial = r\p(r) cos (pz 

 longitudinal = <p(r) sin (pz - 



- a) cos it, 

 a) cos kt, 



<p(r) is found in terms of \p(r) by means of the equations 

 established in Part II. From these equations is deduced a 

 differential equation of the fourth order for ty(r), and for this a 

 solution is obtained containing only positive integral even powers 

 of r. A relation exists, determining all the constants of the 

 solution in terms of the coefficients a and a. 2 of r° and r 2 . In 

 applying this solution to the problem mentioned, terms contain- 

 ing powers of r above the fourth are neglected, and it is shown 

 to what extent the results obtained are approximate. 



On the curved surface, the two conditions that the normal and 

 tangential stresses must vanish lead to the following relation 

 between k and / — 



Here p denotes the density and a the radius of the beam, 

 while M is Young's modulus, and <r the ratio of lateral contrac- 

 tion to longitudinal expansion for terminal traction. This agrees 

 with a result obtained by Lord Rayleigh ("Theory of Sound," 

 vol. i. § 157) on a special hypothesis. 



Proceeding to the terminal conditions, it is shown how / is 

 determined from the conditions at the ends. Since a depends 

 only on the amplitude of the vibrations, we are left with no 

 arbitrary constant undetermined. If the bar be so " fixed " at 

 its ends that the radial motion is unobstructed, this leads to no 

 difficulty, but if an end be " free " a difficulty arises. At such 

 an end the solution requires the existence of a radial stress 

 oc (2* + i) 3 r (a 1 - r' 2 )// 3 , where i is an integer depending on 

 the number of the harmonic of the fundamental note, and / 

 denotes the length of the bar. There will thus be a difference 

 in these cases between the results of experiment and those of 

 the accepted theory, even as amended by Lord Rayleigh. This 

 divergence will increase rapidly with the order of the harmonic, 

 and, though very small for a long thin bar, will increase rapidly 

 as the ratio of the diameter to the length is increased. Since, 

 in dealing with the conditions at the curved surface, terms of 

 the order (ajlf were neglected, the same remarks apply, though 

 to a smaller extent, in the case of the " fixed-fixed" vibrati ons. 



May 31. — "Investigations on the Spectrum of Magnesium. 

 No. II." By Profs. Liveing and Dewar. 



Since our last communication on this subject, we have made 

 many additional observations on the spectrum of magnesium 

 under various circumstances, and have arrived at some new 

 results. Speaking generally, we find that differences of tem- 

 perature, such as we get in the flame of burning magnesium, in 

 the arc, and in the spark, produce less differences in the 

 spectrum than we had before attributed to them. For instance, 

 the lines which previously we had observed only in the spark 

 discharge, we have since found to be developed in the arc also, 

 provided the discharge occur between electrodes of magnesium. 1 

 In making these experiments we used thick electrodes of 

 magnesium, and brought them together inside a glass globe 

 about 6 inches in diameter, fitted with a plate of quartz in front 



1 Compare the appearance of the lines of hydrogen in the arc discharge. 

 Roy. Soc. Proa, vol. xxx. p. 157 ; and vol. xxxv p. 75. 



