Memoir on the Equation the roots of which are the principal moments of inertia of a solid body, and on some other Equations of the same kind. By the same Author.

M. CAUCHY remarks that it has not hitherto been shewn, that the roots of the above equation are possible, except by indirect methods, such as, for instance, the reduction of the cubic equation to a quadratic, by the transformation of coordinates in space: he then proposes a direct demonstration, depending on some theorems enunciated but not demonstrated in this paper.

Memoir on the Movement of a System of Molecules which attract or repel each other at very small distances, and on the Theory of Light. By the same Author.

THE author states, that the equations of motion of such a system of molecules may be integrated by the methods given by himself in the 19th Number of the Journal de l'Ecole Polytechnique, and that these integrals have led him to form the following conclusions:

1st. If in any system of molecules the electricity of the system be equal in every direction, and a vibratory motion be produced at any point, two spherical undulations, of constant, but unequal velocities, will be propagated from that point. The first of these vibrations will cease, when the initial dilatation of the volume ceases; if, then, we suppose the vibrations to have been originally parallel to a given plane, they will continue so.

2d. If the system be such that the elasticity continues the same about an axis parallel to a given line, in all directions perpendicular to the axis, the equations of motion will contain several coefficients depending on the nature of the system; and a relation may be established between their coefficients, such that the propagation of a vibration first produced at any point may give rise to three undulations, each of which coincides with a surface of the second degree. Furthermore, omitting that undulation which disappears with the dilatation of the volume, when the elasticity again becomes equal in every direction, the surfaces of the two remaining undulations will be reduced to a system comprising a sphere and a spheroid, of which the axis of revolution coincides with the diameter of the sphere.

The author observes, that the remarkable coincidence of this result with the theorem of Huyghens on the double refraction of light in crystals having a single axis, is deserving of our attention; and we may hence be induced to conclude, that the equations of the motion of light coincide with those which express the movement of a system of particles very little disturbed from the position of equilibrium.

VOL. I.

MAY, 1831.

2 R

Analytical Demonstration of the Law discovered by M. Savart, relative to the vibrations of Solids and Fluids. By the same Author.

It is here shewn that the equations of motion of an elastic body, the particles of which are very little moved from their natural position, may, by a slight modification, be applied to the demonstration of the above proposition.

The author concludes by observing, that it may be proved in the same manner that if the dimensions of a body increase or diminish in a given ratio, and the initial temperature increase or diminish in the same ratio, the duration of the propagation of heat will vary as the square of that ratio.

Memoir on Torsion and the vibrations of Torsion in a rectangular Rod. By the same Author.

In this paper the author obtains some analytical formulæ, from which he deduces the following results:

i. The angle of torsion of a rectangular rod, fixed at one end and free at the other, when measured in a plane perpendicular to the axis of the rod, is as the distance of that plane from the fixed end, and the moment of the force applied at the free end jointly.

ii. If the transverse section of the rod is variable, but continues similar to itself, the angle of torsion will vary inversely as the square of the area of the section. These results, similar to those obtained by M. Poisson for the torsion of a homogeneous cylindrical rod with a circular base, will hold equally for a circular or prismatic rod on any base.

iii. If one transverse dimension of the rod becomes very small, compared with the other, the angle of torsion will vary inversely as the greater dimension, and the cube of the lesser.

iv. The tones produced by the vibrations of torsion of a rectan gular rod are invariable, so long as the breadth and thickness of the rod are in the same ratio. This is confirmed by the experiments of M. Savart.

v. If one transverse dimension of the rod becomes very small, compared with the other, the lowest tone produced by the vibrations of torsion is directly as the least dimension, and inversely as the area of a transverse section. This is another of the laws to which M. Savart has been led by experiment.

vi. If the elasticity of the rod is the same in every direction, the tones corresponding with the vibrations of torsion will be directly as the product of the two transverse dimensions, and inversely as the sum of their squares.

vii. When the two transverse dimensions are equal, the lowest tone produced by longitudinal vibrations will be to the lowest tone produced by the vibrations of torsion :: 1.9364... : 1.

[To be continued.]

599

FOREIGN AND MISCELLANEOUS INTELLIGENCE.

§ I.-MECHANICAL SCIENCE.

1. STIFFNESS AND STRENGTH OF TIMBER.

In a series of experiments undertaken by Lieutenant T. S. Brown, to ascertain the relative stiffness and strength of different kinds of pine used in building, it was found that the ratios of the stiffness of white pine, spruce, and southern pine, were as the numbers 1, 1·111 and 1807. When the weights producing the flexure were increased, it was found that the failures of the wood began at the top. The upper fibres, for rather less than half the depth of the beam, were gradually crushed and broken off in the bending of the specimen ; and at last, when no more weight could be supported, a fracture suddenly took place, the lower fibres being drawn asunder *.

2. PROPORTION BETWEEN THE METRE AND ENGLISH YARD. M. Francœur, in an elaborate memoir on the proportion between French and English measure, has found that the mètre is equal to 39.37079 English inches, and the English imperial yard equal to 0.91438348-numbers which may be relied on with the utmost confidence t.

3. ON THE VELOCITY OF AN ELASTIC FLUID WHICH FLOWS FROM A RESERVOIR INTO A GASOMETER.

In most books of natural philosophy we find the following proposition stated as if it had been founded on data as certain as the pressure of the atmosphere. 'The flowing of an elastic fluid into a vacuum takes place with the velocity due to the height of a column of fluid of the same density with that contained in the vessel, and whose weight would produce the pressure to which the fluid is submitted.' M. Navier has shewn that this could only be true on the supposition, that the density of the fluid in the tube or orifice through which it flows was the same as that in the reservoir, which, from the diminished pressure, is obviously not the case. The results to which he has arrived will for ever banish this false proposition from books of natural philosophy +.

4. ON THE DISCHARGE OF A JET OF WATER UNDER WATER.

(R. W. Fox, Esq.)

Having observed that a communication of mine On the Discharge of a jet of water under water,' inserted in No. 47, of the Philosophical Magazine, has been noticed in the last number of the Journal of the Royal Institution, I will take this opportunity of mentioning,

*Silliman's Journal, vol. xix. p. 292.

Mem. de l'Acad. des Sciences, tome x. p. 50.

Ibid.

that where a jet of water is discharged under mercury, the results are the same, under a given force, as when it takes place in water, or air, the quantity discharged being in all cases the same, in the same time.

Hence, it appears that the force with which a moving or spouting fluid recoils is not affected by the surrounding medium, however rare or dense it may be: and thus we may understand why the attempts, which have been made to propel vessels, by forcing water through them against water, have not proved advantageous.

The well known fact that large rivers penetrate, in a direct course, far into the ocean, notwithstanding its agitation by tides and currents, is somewhat analogous; and were it not for this remarkable degree of mobility in water, the sediment, which is now mostly deposited at a considerable distance in the sea, would accumulate near the mouths of rivers, and tend to divert them from their course.

Whilst making my experiments on the jet of water, I noticed that when sand was dropped into the water near the orifice from which the jet issued, it was drawn laterally toward the hole, till it distinctly appeared to enter it, but it was in fact only an optical deception, the grains of sand being carried away by the jet as soon as they came in contact with it, with such great velocity as to be perfectly invisible. 5. OPTICAL DECEPTION UPON THE LIVERPOOL AND MANCHESTER RAIL ROAD.

This rail road consists of two lines of rails, so that in looking from the carriage window of one line, the other is seen, and presents the following somewhat remarkable appearances. While travelling at the rate of from 12 to 15 miles per hour, the rail, together with the roadway, the banks, and other objects, appear, as they do from the window of a stage-coach, to recede, or move in a direction the reverse of that in which the carriage is moving. But when the speed increases to twenty-four or thirty miles in the hour, the rails no longer seem to recede, but to move with the carriage, as if they were running along the road at the same rate as the spectator. These different appearances, accompanying the different speeds, are explained without difficulty; they depend on the facts which are familiar to every one who has caused a fluted pencil case, having a plain slider, to revolve in the hand. The case is obviously seen to move, but the slider seems stationary, because, being plain, it presents at every period of its revolution precisely the same appearance to the eye. If the iron rails appeared always to move along the road, the explanation of the phenomenon would already have been given. The varying effect produced by varying speeds depends on the circumstance that the iron rails are not, like the slider of the pencil case, quite plain, but have slight irregularities occurring at short intervals, which with a moderate velocity are visible, and give to the rail a receding appearance; but when the velocity becomes doubled, the impression on the retina produced by one irregularity is not effaced till it is succeeded by others, so nearly similar, that the appearance of the

rail resembles that which would be given by one without any irregularities at all. The varieties of form and colour in the road and banks are too great, and separated by too long intervals, to be effaced by any possible speed, and therefore they continue to recede from the carriage after the rail seems to have changed the direction of its A. A.

motion.

6. A BAROMETER OF A NEW CONSTRUCTION.-(Proposed by

M. Kupffer.)

The great danger of particles of air entering into the vacuum of a barometer while in use, and the difficulty of effectually expelling such air by boiling the mercury, have induced M. Kupffer to propose a construction of this instrument by which the error arising from this source, instead of being removed, may be taken into account. The cistern of the barometer is a hollow iron cylinder; the bottom of which can be raised and lowered at pleasure by means of a screw. Besides the ordinary tube of the barometer, another smaller tube, like the short arm of a syphon, proceeds from, and communicates with the cistern, which is quite full of mercury. The height of the mercury in the two tubes, the difference of which is the height of the barometer, may be changed at pleasure by means of the above screw; and the difference of level is measured by a scale, one extremity of which, a steel point, reaches down into the shorter tube, so that it may be brought into contact with the surface of the mercury in it. By the action of the same screw, the space above the mercury in the long tube, which ought to be a vacuum, may consequently be reduced at pleasure to any degree. Now, if we suppose that this space contains some air, let the height of the barometer, which was observed while the space was e, be A, and let the pressure of the air in the tube at the same moment bep. After reducing the space to the smaller quantity c', let the height of the barometrical column be B, and the corresponding pressure of the air = p'; then will be, by the law of Mariotte, P = and the p' e

corrected height of the barometer

Р

e'

A+ p = B+ p'; whence
A-B

we find the correction to be applied to A, viz. Р =

e 1

Ifc=2e',

or if the space at the second observation is exactly half of what it was at the first observation, we shall have p = A B. Thus the difference of the observed heights will, in that case, be exactly the correction to be applied to the first height for the error arising from the action of the air *.

7. OCCULTATION.

M. Tarkhanoff, the astronomer of an imperial Russian expedition for • Petersburgh Transactions, 1830.