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subject to the i (21+1) equations
AB+ A' B'+A" B"=0
gives, from any one canonic group, another indeterminately. To find the degree of indeterminateness, let absolute magnitudes in canonical forms be ruled by the conditions
SH H’ do=fH?do=..=1,
SH” do=sS H” do =..=1, we have therefore
B + B'?+B" +...=1, as in the ordinary transformation of rectangular axes in three dimensions. These 2i+l equations with the i (2i+1) previous, amount in all to (i+1) (2i+1) equations of condition among (2i+1)* coefficients, leaving i (2i+l) independent variables.
The only canonical form hitherto generally recognized is that of Laplace; consisting* of 2i+1 polar harmonics, of which 1 is zonal, 2 (i-1) are tessaral, and 2 are sectorial. In the discussion which followed Mr. Clifford's paper on this form, I remarked that it seemed to be a singular case of the general canonic; notably singular in this respect, that for any one of its constituents the nodal cone consists of circular cones having a common axis and planes through this axis. The nodal cone of any spherical harmonic of degree i is an algebraic surface of degree 2i+1, and I proposed the question, can canonical forms not be found in which the nodal cone of each constituent is not resolvable into circular cones and planes ? This question is answered by the present communication.
[A diagram was roughly sketched on the board, to illustrate the nodal cone of a harmonic differing infinitely little from a tessaral harmonic; which, with 2i others differing infinitely little from the other 21–3 tessaral, the two sectorial, and the zonal, constituting the polar canonic, would constitute a generalized canonic.]
Account of Experiments upon the Resistance of Air to the Motion of Vortex
rings. By ROBERT STAWELL Ball, A.M., Professor of Applied Mathematics and Mechanism, Royal College of Science for Ireland.
The experiments, of which the following is an abstract, were carried out with the aid of a grant from the Royal Irish Academy. A paper containing the results has been laid before the Academy.
The author proposed to bring this subject before the Association in order to elicit discussion. He would greatly value any suggestions as to the direction in which future experimental researches would be likely to prove fruitful. Such suggestions, though acceptable from all sources, would come with peculiar usefulness from those who are conversant with the profound hydrodynamical problems of vortex motion,
A brief account of one series of the experiments, and a Table embodying them, will be given.
Air-rings, 9 inches in diameter, were projected from a cubical box, each edge of which is 2 feet. The use of this box was suggested by Professor Tait (see a
* Thomson and Tait's 'Natural Philosophy,' $ 781.
paper by Sir William Thomson, Phil. Mag. July 1867; also a paper by the Author, Phil. Mag. July 1868). The blows were delivered by means of a pendulum called the striker, which, falling from a constant height, ensured that the rings were projected with a constant velocity. In the experiments described in the present series, this velocity was somewhat over 10 feet per second. The pendulum was released to deliver the blow from a pair of forceps, each jaw of which was in connexion with a pole of a battery. After the ring had traversed a range varied from 2 inches to 20 feet, it impinged upon a target. The blows upon the target closed the circuit, which had been opened at the release of the striker. An electric chronoscope (devised, it is believed, by Wheatstone) measured the interval of time between the release of the striker and the impact upon the target.
The target was placed successively at distances of 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 feet from the orifice of the box. Not less than ten observations of the time were taken at each range. The probable error of the mean time at each range is in every case less than 1 per cent. of the whole amount. A special series of experiments, which need not be described, determined the value of the chronoscope readings in seconds.
The observations are next represented in a curve, of which the abscissæ are the ranges, and the ordinates the corresponding mean chronoscope readings. By drawing tangents to this curve, the velocity of the ring at its different points is approximately found.
A second projection is made in which the abscissæ are the ranges and the ordinates are the velocities; the points thus determined are approximately in a straight line.
It follows that the rings are retarded as if acted upon by a force proportional to the velocity, and an approximate value of the numerical coefficient becomes known.
A more accurate value having been determined by the method of least squares, the results are embodied in the following Table (p. 28), of which a description is first given. The Roman letters refer to the several columns of the Table.
I. contains a series of numbers for convenience of reference.
II. It was found that the motion of the ring in the immediate vicinity of the box was influenced by some disturbing element. The zero of range was therefore taken at a point 4 feet distant from the orifice. This column contains the ranges.
III. The interval between the release of the striker and the arrival of the ring at a point 4 feet from the orifice is 6.5 chronoscopic units, or about 0.93 second. This constant must be subtracted from the mean readings of the time, in order to reduce the zero epoch to the instant when the ring is 4 feet from the orifice. This column contains the mean readings of the chronoscope corrected by this amount.
IV. When the ranges are taken as abscissæ, and the corresponding times as ordinates, it is found that a curve can be drawn through or near all the points thus produced. To identify the points with the curve, small corrections are in some cases required. These corrections are shown in column IV. In the case of experiment 5 the correction amounts to 0:7; this is about 0·09 second. The magnitude of this error appears to show that some derangement, owing possibly to a current of air or other source of irregularity, has vitiated this result
. For the sake of uniformity, however, the corrected value has been retained.
V. This column merely contains the corrected means, as read off upon the curve determined by the points, VI. The value of the chronoscope unit after the first few revolutions is
0.1288 second, with a probable error of 0.0002 second. By means of this factor the corrected means in column V. are evaluated in seconds in column VI.
VII. This column contains the time calculated on the hypothesis that the rings are retarded as if acted upon by a force proportional to the velocity, the coefficients being determined by the method of least squares ; the formula is
t=9-016-6-25 log (27:7-8).
Table of Experiments, showing the retardation which a Vortex-ring of air experiences when moving through air at the same temperature and pressure. The vortex-ring is 9 inches in diameter, and has an initial velocity of 10-2 feet per second. The retarding force is proportional to the velocity; and after 2.24 seconds the ring has moved 16 feet, and its velocity is reduced to 4:3 feet
Distance of Mean corre-Correction Corrected
target from Reference
sponding deduced by value of Equivalent number. orifice, reading of graphical
time, in minus chronoscope, construc- chronosco- seconds. 4 feet, minus 6.5.
tion. pic reading.
Calculated time by formula t=9'016 -6•25 log (277–8).
Retarding Approximate True velocity, between
calculated by calculated by
VIII. This column shows that the difference between the corrected mean time and the calculated time in no case exceeds
0-01 second. IX. The approximate velocities, deduced by drawing tangents to the curve. X. The true velocities, calculated from the formula
dt XI. The retarding force, calculated by
Experiments on Vortex-rings in Liquids. By H. Deacon.
On Units of Force and Energy. By Professor J. D. EVERETT, F.R.S.E. The object of the paper was to urge the necessity of giving names to absolute units of force and energy, that is, units not varying with locality, like the gravitation units vulgarly employed (pound, foot-pound, &c.), but defined by reference to specified units of length, mass, and time, according to the condition that unit force acting on unit mass produces unit acceleration.
The author proposed that the units of force and of energy (or of work), thus related to the metre, gramme, and second, be called respectively the dyne and the pone (8úvapıs, Tóvos), and that the names kilodyne, megadyne, kilopone, megapone be employed to denote a thousand and a million dynes and pones. The megadyne and megapone will thus be the units of force and energy related to the metre, the tonne, and the second.
He also proposed that the units of force and energy related to the foot, the pound, and the second be called respectively the kinit and the erg *.
On the Corrosion of Copper Plates by Nitrate of Silver.
By J. H. GLADSTONE, F.R.S., and ALFRED TRIBE, F.C.S. In some recent experiments in chemical dynamics, the authors had occasion to study the action of nitrate of silver on copper plates in various positions. They observed that when the plate was vertical there was rather more corrosion at the bottom than at the top. This is easily accounted for by the upward current, which flows along the surface of the deposited crystals, and which necessitates a movement of the nitrate-of-silver solution towards the copper plate especially impinging, on the lower part. It was also found that when the copper plate was varnished on one side it produced rather more than half the previous decomposition, and was most corroded at the edges of the varnish. By making patterns with the varnish, this edge action became very evident. This was explained by the fact that the long crystals of silver growing out from the copper at the borders can spread their branches into the open space at the side, and so draw their supply from a larger mass of solution than the crystals in the middle can do ; and increased crystallization of silver means increased solution of copper. This was proved by making the varnish a perpendicular wall instead of a thin layer, when the greater corrosion was not obtained. In a plate completely surrounded with liquid, the greatest growth of crystals is also evidently from the angles. It was likewise observed that if a vertical plate be immersed, the lower part in nitrate of copper, and the upper part in nitrate of silver, there is greater corrosion about the point of junction. This was attributed to the greater conduction of the stronger liquid.
Some Remarks on Physics. By M. JANSSEN.
* Since the reading of the paper, a Committee has been appointed by the Association “to frame a nomenclature of absolute units of force and energy."
On Democritus and Lucretius, a Question of Priority in the Kinetical Theory
of Matter. By T. M. LINDSAY and W. R. SMITH, Physicists who have recently called attention to the anticipation of modern doctrines as to the ultimate nature of matter by the ancient atomists, have looked too exclusively to Epicurus and his expositor Lucretius, to the neglect of Democritus and Leucippus. Democritus had no such expositor as Lucretius, but his main views are accessible in the fragments collected by Mullach, and in the wellknown references of Aristotle, Simplicius, and Laertius. With the help of these sources, the paper sketches the main features of the earliest atomic theories. The following are leading points :
Democritus and Leucippus trace the variety of phenomena to three primitive differences in the ultimate elements of nature, viz. dífferences (1) in Figure, oxiua, as between A and N; (2) in Order, táśls, as AN, NA; (3) in Position, béous, as Z, N [Arist. Met. A. 4). From the motion in racuo of atoms with these primary differences, the whole variety of nature is deduced, generation and corruption being merely syncretion and division (cúykplois, diákplois) (Ar. De Gen. et Cor. i. 8, i. 2, Phys. viii. 9]. All atoms have the same density and the same opun tîs popas (specific gravity ?) [Ar. De Cælo, i. 7, Theophrastus De Sensu, 71). Hence all tend to fall in one downward direction * ; but
being ignorant of the law of inertia, Democritus supposes that the larger atoms fall faster, impinge on lighter particles, and produce a vortex motion (diun). In this vortex similars come together and cohere, lighter particles go to the surface, and at length worlds (kóopol) are generated (Diog. Laertius, ix. 31). Epicurus differs from Democritus mainly by maintaining that all atoms have equal and invariable downward velocities, and come into collision only by fortuitous automatic deflection from the line of fall. The first half of this theory looks like the first law of motion, but is really as far from being in harmony with the laws of acceleration and other known truths as the earlier view. As physicists, therefore, Epicurus and Lucretius made no advance on Democritus, while by mixing up with legitimate physical speculation the incongruent metaphysical notion of chance (not the mathematical notion of chance, which plays a part in the modern kinetic theory of gases), they produced that hybrid of physics and metaphysics, a materialistic philosophy. It was by adopting the Epicurean doctrine of chance that Gassendi, the first of modern atomists, became also the father of modern materialism.
Speculations on the Continuity of the Fluid State of Matter, and on Relations
between the Gaseous, the Liquid, and the Solid States. By Prof. JAMES THOMSON, LL.D.
Through the recent discovery of Dr. Andrews on the relations between different states of fluid matter, a dificulty in the application of our old ordinary language has arisen. He has shown thě existence of continuity between what is ordinarily called the liquid state and what is ordinarily called the gaseous state of matter. He has shown that the ordinary gaseous and ordinary liquid states are only widely separated forms of the same condition of matter, and may be made to pass into one another by a course of continuous physical changes presenting nowhere any interruption or breach of continuity. If, now, there be no distinction between the liquid and gaseous states, is there any meaning still to be attributed to those two old names, or ought they to be abandoned, and the single name the fluid state to be substituted for tħem both? The answer must be that in speaking of the whole continuous state we have now to call it simply the fluid state ; but that there are two regions or parts of it, meeting one another sharply in one way, and merging gradually into one another in a different way, to which the names liquid and gas are still to be applied. We can have a substance existing in two fluid states different in density and other properties, while the temperature and pressure are the same in both: and we may then find that an introduction or abstraction of heat without change of temperature or of pressure will effect the change from the one state to the other, and that the
* Cf. the argument in Zeller, Phil, der Griechen, i. 913, ff.