which is a chisel equal in breadth to the length of the intended teeth of the comb, is brought down with sufficient force to cut completely through the horn or tortoiseshell. As the teeth are not required to be perfectly parallel, but in a slight degree wedge-shaped, the successive cuts of the chisel must incline a little obliquely, first in one direction, and then in the other, for which provision is made by an arrangement for alternately changing the position of the cutter during its descent. A pair of very narrow chisels mounted at right angles with, and at the ends of, the long cutter, serve, by their alternate descent, to connect each pair of cuts at their converging extremities, so as to detach the ends of the teeth. In Mr. Rogers's machine for the same purpose, everything is effected by the continuous turning of a winch, which, by means of a crank, works a double cutter, the two chisels of which are capable of adjustment according to the size of the teeth, while the screw for moving the bed or carriage is turned by means of a cogwheel upon its axis, working into another on the axis of the winch. The extent of motion between each operation of the cutter is regulated by varying the relative sizes of these cog-wheels.

Among recent novelties in comb-making is Mr. Griffith's curious patent, in 1852, for "galvanic combs," of which the teeth are alternately of copper and zinc, while the handle is hollowed into a chamber for containing a roll of flannel moistened in acid solution. The inventor seems to expect a beneficial galvanic action by combing the hair with this apparatus. A more promising novelty is the application of the very tough substance called vulcanised india-rubber [CAOUTCHOUC MANUFACTURES] as a material for combs.

COMBAT, SINGLE. [DUEL.]

COMBINATION LAWS. The laws known by this name were repealed in 1824. Till then any combination of any two or more masters, or of any two or more workmen, to lower or raise wages, or to increase or diminish the number of hours of work, or quantity of work to be done, was punishable at common law as a misdemeanour and there were also thirty-five statutes in existence, most of them applying to particular trades, prohibiting combinations of workmen against masters. The Act passed in 1824 (5 Geo. IV. c. 95) repealed all the statute and common law against combinations of masters and of workmen; provided a summary mode of conviction, and a punishment not exceeding two months' imprisonment for violent interference with workmen or masters, and for combinations for violent interference; and contained a proviso with regard to combinations for violent interference, that no law in force with regard to them should be altered or affected by the Act. But all the common law against combinations being repealed by the Act, this proviso was considered as of no force; and the Act also went beyond the intentions of the framers in legalising combinations unattended with violence for the purpose of controlling masters in the mode of carrying on their trades and manufactures, as well as peaceable combinations to procure advance of wages or reduction of hours of work. The Act was passed after an inquiry into the subject by a committee presided over by Mr. Hume, which reported to the House the following among other resolutions :—

"That the masters have often united and combined to lower the rates of their workmen's wages, as well as to resist a demand for an increase, and to regulate their hours of working, and sometimes to discharge their workmen who would not consent to the conditions offered to them; which have been followed by suspension of work, riotous proceedings, and acts of violence.

"That prosecutions have frequently been carried on under the statute and the common law against the workmen, and many of them have suffered different periods of imprisonment for combining and conspiring to raise their wages, or to resist their reduction, and to regulate their hours of working.

That several instances have been stated to the committee of prosecutions against masters for combining to lower wages, and to regulate the hours of working; but no instance has been adduced of any master having been punished for that offence.

"That it is the opinion of this committee that masters and workmen should be free from such restrictions as regard the rate of wages and the hours of working, and be left at perfect liberty to make such agreements as they may mutually think proper.

That therefore the statute laws which interfere in these particulars between masters and workmen should be repealed; and also that the common law, under which a peaceable meeting of masters or workmen may be prosecuted as a conspiracy, should be altered.”

Immediately after the passing of this Act a number of widely organised and formidable combinations arose in various trades and manufactures for the purpose of controlling the masters as to the way in which they should conduct their business; and the extent to which the Act had repealed the common law being doubtful, and the Act having clearly gone beyond the resolutions on which it was grounded in legalising combinations, Mr. Huskisson, then President of the Board of Trade, moved early in the session of 1825 for a committee to consider the effects of the Act 5 Geo. IV. c. 95; and a committee was appointed with Mr. (afterwards Lord) Wallace, then Vice-President of the Board of Trade, for its chairman. This committee recommended the repeal of the Act of the previous session, and the enactment of another; and in consequence of their recommendation the 6 Geo. IV. c. 129, was passed, which is the Act now in force relative to combinations.

This Act repealed the 5 Geo. IV. c. 95, and all the statutes which that Act had repealed. It relieved from all prosecution and punishment persons meeting solely to consult upon rate of wages or hours of work, or entering into any agreement, verbal or written, on these points. And it provided a punishment of not more than three months imprisonment, with or without hard labour, for any one using violence, or threats to make a workman leave his hiring, or return work unfinished, or refuse to accept work, or belong to any club, or contribute to a common fund, or pay any fine for not belonging to a club, or contributing to a common fund, or refusing to conform to any rules made for advance of wages or lessening of the hours of work, or regulations of the mode of carrying on any business, and for any one using violence to make any master alter his mode of carrying on his business. By the Act 6 Geo. IV. c. 129, therefore combinations of masters and workmen to settle as to rate of wages and hours of labour are made legal and freed from all punishment; but the common law remains as it was as to combinations for otherwise controlling masters. Although, therefore, workmen may conspire together not themselves to work under certain wages, they must carry out their object by lawful means, and cannot intimidate or prevent masters from employing, or workmen from taking employment, at any wages those other workmen may agree for. (Reg. v. Rowlands, 2 Den. C. C. 364.)

By 9 Geo. IV. c. 31, assaults in pursuance of a combination to raise the rate of wages are made punishable by imprisonment and hard labour.

COMBINATIONS AND PERMUTATIONS. By the word combination is usually meant any selection which can be made out of a number of different objects without reference to the order in which they are placed; while by a permutation is meant a combination in which different orders of position are to be considered as constituting a specific difference. Thus abcd, acbd, da cb, are all the same combination of four out of the alphabet, but different permutations of four. The investigation of questions relating to combinations, &c., is the principal mathematical part of the theory of probabilities, and was first considered in detail, with reference to that science, by James Bernouilli and Montmort (see Library of Useful Knowledge: Probability'); but the common rules had previously found their way into arithmetical treatises. The enormous number of different arrangements of which objects are susceptible, even when their number is not large, drew early attention to the subject. We shall give some of the most simple rules, and a help to calculation for high numbers.

I. The number of permutations having x in each, which can be made out of X things, is the product of x terms of the series, X, X-1, X2, X-3, &c.

Thus, out of 10 things, there are 10 x 9, or 90 permutations of two; 10 x 9 x 8, or 720 permutations of three; 10 x 9 x 8 x 7, or 5040 permutations of four; and so on. Finally, the number of different arrangements which the whole ten will admit of, say the number of changes which can be rung on ten bells, is

10. 9. 8. 7. 6. 5. 4. 3. 2. 1. or 3,628,800.

II. When the whole number of things, X, contains a which are alike of one sort, b which are alike of another sort, &c., the total numdown to 1, but that product divided by the product of 1, 2, 3. ber of arrangements of the whole is not the product of X, X - 1, &c., up to a, then by that of 1, 2, 3. up to b, &c. This result can be most denominators.

...

easily formed by striking out common factors from the numerators and

α α

C.

being required, no simple rule can be given, but each case must be
III. In the last case, the number of permutations of x out of X
formed out of
solved by itself. For instance, how many permutations of three can be
a b b
(1.) All being different, 3. 2. 1. or 6. (2.) Where a is repeated
twice, we have 6. (3.) Where a is repeated three times, one only.
(4.) Where b is repeated twice, we have 6. In all, 19.
IV. The number of combinations of x things out of X, all
different, is

prod. of x terms of X, X-1, X — 2, &c.
divided by

prod. of x terms of 1, 2, 3, &c.

Thus out of 10 things, the number of combinations of four is 10. 9. 8. 7. divided by 1. 2. 3. 4, or 210. The best way of arriving at this result is by destroying common factors, which shows it to be 5. 3. 2. 7. Observe also that we may shorten this process, when x is greater than the half of X, by finding out, not how many selections can be taken, but how many remainders can be left. Thus the number of combina tions of 25 out of 30, is the same as the number of combinations of 5, for 25 can only be taken in as many ways as 5 can be left. V. The number of combinations of x things out of X, any repetition being allowed, is

prod. of x terms of X, X + 1, X + 2, &c.
divided by

prod. of X terms of 1, 2, 3, &c.

VI. The number of ways in which n places may be filled up from x

letters, allowing any letter to be repeated in all or or any of the places, is a", or the product of x, x, x, (n factors in all). This is the number of permutations of n out of x, allowing repetition. VII. The number of ways in which n different letters can be distributed into boxes, all possible modes of distribution being equally allowable, is x".

VIII. The total number of combinations of all sorts out of x things, from one at a time up to all together, both inclusive, is 2*, or 2. 2. 2. (x factors in all) diminished by 1. Thus out of 4 things, there are 21, or 15 different sections: they are

abcd, bed, a cd, abd, abc, ab, ac, ad, bc, bd, cd,

a, b, c, d.

Among the curiosities of this subject, it will suffice to mention the following: The number of all possible arrangements of letters, repeated or not, and capable of being pronounced or not, up to words of 24 letters, is of the following order of magnitude: Take a million of millions; repeat it a million of million times: the result is between 1391 and 1392 millions of such numbers. As an instance of the manner in which the dropping of consonants and confusion of vowels may permit possible alterations of spelling, M. de Mairan computed that the word Hainaut might be spelt in 2304 different ways, so as to be pronounced in the same way by as many different Frenchimen, or very nearly so.

....

combustion of each other. In like manner a jet of mercury vapour is combustible in an atmosphere of chlorine, and a jet of chlorine is combustible in an atmosphere of mercury. In these and other similar instances the chemical action between the two bodies at their line of contact with each other, is sufficiently intense to produce light and heat, and consequently it is at that line that the phenomenon of combustion ensues.

Spontaneous combustion is combustion that is set up between two bodies at common temperatures, without any application of artificial heat. Thus, the burning of arsenic and of antimony in chlorine, previously referred to, are examples of spontaneous combustion. In kindling a lucifer match friction is necessary to produce a temperature at which the exposed phosphorus will ignite; but a piece of phosphorus, exposed to the direct rays of the sun on a warm day, will inflame spontaneously. Tow or cotton waste, moistened with oil and exposed to the air, frequently undergoes spontaneous combustion, on account of the attenuated state of the oil produced by the fibre being a condition favourable to the chemical action of the oxygen of the air upon it; on this account great care should be taken in factories where oily machinery is cleaned with cotton waste, to prevent the accumulation of materials of that description.

which is a little too small, but the error is only about the 12r-th part of the whole less than 1 per cent. even when x is so low as 10. The expression can easily be calculated by logarithms. Tables of the logarithms of this product will be found at the end of the article Theory of Probabilities,' in the Encyclopædia Metropolitana.' For an instance of the computation, see the Library of Useful Knowledge: 'Examples of Arithmetic,' &c., p 45.

COMBINING PROPORTION. [ATOMIC THEORY.]
COMBINING VOLUME. [ATOMIC VOLUME.]

COMBUSTIBLE. In its more restricted and usual sense, this term signifies a body which is capable of combining with oxygen, with the evolution of so much heat as to become luminous or incandescent. [COMBUSTION.]

COMBUSTION is a term usually restricted to describe the phenomenon that ensues when chemical action is sufficiently intense to produce light and heat. The burning of coal, wood, paper, candles, oil, or coal-gas are familiar illustrations of combustion. Less common, but more brilliant, instances of combustion are seen in the explosion of gunpowder, or fireworks, or in the burning of steel-wire, charcoal, or phosphorus in oxygen gas.

In the examples of combustion above alluded to, the action lies between the burning body on the one hand, and pure or diluted oxygen on the other; and inasmuch as our world is enveloped in an atmosphere of which the most important constituent is oxygen, it follows that all ordinary instances of combustion are owing to the rapid oxidisation of bodies at a high temperature. It would be wrong, however, to suppose that the word combustion expresses no other actions than those indicated. Many substances burn equally well in atmospheres from which oxygen is excluded altogether, and in some cases even burn more readily than they would under similar circumstances in pure oxygen. For instance, when the metals arsenic or antimony are finely powdered and thrown into an atmosphere of chlorine, they instantly ignite, and burn with evolution of light and heat; in fact, literally undergo combustion.

Combustibles, and supporters of combustion or non-combustibles, are terms used to designate two distinct classes of substances. Air, oxygen, chlorine, &c., are non-combustible, that is, in the common acceptation of the word; they do not burn, but they support the combustion of other substances, such as wood, coal, &c., which latter are called combustibles. The phraseology is, however, purely conventional, and only applicable so long as the circumstances under which it is applied remain the same. For instance, common coal-gas burns in atmospheric air, and under these circumstances the gas is called the combustible and the air the non-combustible or supporter of combustion. But change the conditions, fill a jar with coal-gas, introduce a jet of common air and ignite the latter, perfect combustion will then go on at the jet: the air may now with equal propriety be said to be the combustible and the gas the non-combustible, for the gas just as much supports the combustion of the jet of air as in the former case the air supported the combustion of the jet of gas; in fact, both are equally combustible, and both equally support the

The term combustion is sometimes used to describe certain chemical actions in which heat is evolved but no light. Thus the heat of the body is sometimes spoken of as being caused by the combustion of the carbon of the blood with the oxygen of the air. Occasionally, the word is used to denote particular chemical actions where not only no light but even no sensible heat is evolved; thus, the decay of animal and vegetable matter is said to be due to slow combustion. It is, however, far more convenient to speak of such as phenomena of oxidation, and restrict the term combustion to the meaning given to it at the commencement of this article. COMEDY. [DRAMA.]

COMENAMIC ACID. [MECONIC ACID.]
COMENIC ACID. [MECONIC ACID.]
COMETARY BODIES. [COMETS.]

COMETS. This term has been applied to bodies of a nebulous aspect which occasionally appear in the heavens, accompanied in the more conspicuous cases by a long train of light called the tail. The principal part of a comet's structure, in contradistinction to the lastmentioned appendage, is denominated the head. The outline of the head is hazy and ill defined; hence the origin of the term comet (Kounτns, from kóun, hair). The head gradually increases in brightness towards the centre, where it assumes a planetary aspect. In some instances it exhibits a small bright central point, bearing a resemblance to a star. This point is called the nucleus. The head of a bright comet is usually shrouded in a paraboloidal envelope of light, the prolongation of which forms the tail. The tail is turned in the direction opposite to the region in which the sun is situate. Its outline is gently curved, being convex on the side towards which the comet is travelling. This remarkable appendage frequently extends over a considerable arc of the heavens, imparting a grand and mysterious aspect to the object with which it is connected. Nor is the length of the tail merely apparent; on the contrary, it not unfrequently extends to an enormous distance in space. Thus, for example, the tail of the great comet of 1843 attained a maximum length of 150 millions of miles; while the tail of the great comet of 1858, at the time of its greatest development, did not certainly fall short of 50 millions of miles in length.

The tail, however, must not be considered as forming an essential part of the structure of a comet. Multitudes of bodies of this class have been observed in modern times, which exhibited merely a round nebulous mass of light without the slightest vestige of a tail. Such bodies are generally visible only by the aid of the telescope; but even in some instances of comets of conspicuous magnitude, no trace of a tail has been discovered. For example, the comet of 1585, observed by Tycho Brahé, is said to have exhibited neither tail nor coma, but appeared perfectly round like a planet. ("Planè rotunda extitit; nec ullam caudam aut barbam in unam magis quam in aliam partem portendebat." *) Cassini relates a similar fact with respect to the comets of 1665 and 1682.

The appearance of a great comet in the heavens has in all ages strongly attracted the attention of mankind. During the earlier periods of history, bodies of this class were generally contemplated with superstitious dread as omens of divine displeasure, and were regarded

Epist. ad Landgrav. p. 13.

891 A.D. Contemporary European writers allude to a great comet which appeared in this year. In the Chinese annals, wherein allusion to it is also to be found, it is stated that the tail was 100° in length. 1106. A great comet appeared, which was visible over all Europe. The tail is said to have resembled a fiery beam. According to Matthew Paris, it was visible in the day-time.

1264. This year was distinguished by the apparition of a magnificent comet, which is alluded to by several European writers, and is also mentioned in the Chinese annals. The tail is said to have attained a length of 100°.

1402. Two comets of extraordinary splendour appeared in the course of this year. The first became visible in the beginning of spring, and towards the close of March was so bright as to be visible in the day-time. The second comet of the year was not less conspicuous than the first, if we are to believe the statements of contemporary writers. The tail is said to have extended from the horizon to the zenith. It is stated, also, as in the case of the first comet, to have been visible in full daylight.

1456. A magnificent comet was visible throughout all Europe. The tail is said to have been 60° in length. The apparition of the phenomenon excited universal terror, in consequence of its being simultaneous with the capture of Constantinople by the Turks. With the view of averting the evil influence of its presence, Pope Calixtus II. ordered prayers to be offered up in all the Western churches; he also, in a famous bull, anathematised at once the Turks and the comet. It has been satisfactorily established in modern times that this was one of the early apparitions of Halley's comet.

1472. The comet of this year was undoubtedly the most splendid of the century. Towards the end of January it was visible in full daylight. In Europe Regiomantanus observed it. In China its successive positions with respect to the stars were also carefully recorded.

1531. An early apparition of Halley's comet. Observed in Europe by Peter Apian, at Ingoldstadt. An account of this apparition is also to be found in the Chinese annals.

1532. A comet appeared this year, which is stated by Cardan to have been visible in full sunshine. 1556. Apparition of a great comet, which has been supposed by some astronomers to be identical with the comet of 1264.

1577. The comet of this year is memorable in history from having furnished the data which enabled Tycho Brahé to demonstrate that the regions traversed by cometary bodies in general lie beyond the moon's orbit. 1607. An apparition of Halley's comet. The phenomenon was observed on the Continent by Kepler and Longomontanus, and in En land by the celebrated mathematician, Hariot. The head is said to have equalled in size the planet Jupiter, but to have shone by a pale and watery light. The tail, which was of a very conspicuous brightness, was about 7° long.

1618. The third comet of this year was one of the most splendid of which history makes mention. Longomontanus states that the tail was 100° long.

1652. Apparition of a conspicuous comet, which is minutely described by Hevelius. 1664-5-8. Each of these years was distinguished by the apparition of a comet of considerable brightness.

1680. The comet of this year is, for several reasons one of the most remarkable of ancient or modern times. It was first seen by Godfrey Kirch, at Coburg in Saxony, on the 14th of November. After its passare of the perihelion on the 20th of December, it shone with great splendour, the tail appearing in some places to extend over an arc of 90. This comet approached nearer the sun than any other comet recorded in history, with the exception of the great comet of 1843. It has been already stated that the observations of this comet furnished the data by means of which Newton was enabled to demonstrate that the orbits of comets are conic sections, having the sun situate in their common focus.

1744. This was the most brilliant comet of the 18th century. It was discovered at Haarlem by Klinkenberg, on the 9th of December, 1743. On the 7th of February the tail was 20° in apparent length. On the 1st of March, when the comet passed the perihelion, it was seen in full daylight. Remarkable physical changes were observed to occur in the head of this comet, on the occasion of its approach to the perihelion. 1759. An apparition of Halley's comet.

1769. This comet is memorable for the immense tail by which it was accompanied. Its passage of the perihelion took place on the 8th of October. On the 10th of September its tail appeared at Paris to be

ARTS AND SCI. DIV. VOL. III.

60°. According to Pingré, the apparent length of the tail in tropical countries measured 97°. 1807. The comet of this year was very conspicuous to the naked eye. It was first discovered at Castro Giovanni in Italy, on the 9th of September, by Parisi, an Augustine monk. The passage of the perihelion occurred on the 19th of the same month. This comet was carefully observed by Sir William Herschel, who, in a paper published in the 'Philosophical Transactions of the Royal Society' for 1808, has recorded many interesting facts respecting it.

1811. The first comet of this year is in many respects one of the most remarkable of modern times. It was discove ed. by M. Flaguerges, at Viviers, on the 26th of March. The passage of the perihelion took place on the 12th of September. From that time till the end of the year it formed a very conspicuous object in the heavens, the effect being enhanced by the circumstance of its apparent path lying so near the North Pole that it always remained above the horizon. It was last seen in Siberia, by Wisniewski, a Russian astronomer, on the 17th of August, 1812.

1835. An apparition of Halley's comet.

1843. One of the most splendid comets recorded in history. It was seen with the naked eye, close to the sun, in Italy, the Cape of Good Hope, and America, on the 28th of February, the day of its passage of the perihelion. In some places the tail was observed to extend over an arc of 65°. It generally disappeared from observation about the beginning of April. This comet is remarkable for having approached nearer the sun than any other comet of modern times.

1853. A very fine comet appeared in the autumn of this year. Throughout Europe it was distinctly visible to the naked eye shortly after sunset.

1858. The comet discovered by Donati on the 4th of June in this year, is one of the most splendid of which history makes mention. It first became generally visible to the naked eye on the 5th of September. The passage of the perihelion took place on the 30th of that month. The comet attained its greatest slendour about the 10th of October. The tail then appeared to extend over an arc of about 40°. The comet ceased to be visible in Europe about the 20th of October, but it continued to be observed by Mr. Maclear at the Cape of Good Hope till the beginning of March in the following year.

Theory of the Movements of Comets.

The theory of the movements of comets resolves itself into two great subjects of research. One of these relates to the determination of the orbit of a comet from a definite number of observed positions, supposing it to revolve in a conic section around the sun; the other takes cognisance of the effects of planetary perturbation upon its motion. Newton's method for determining the elements of a comet's orbit was founded on the hypothesis of its revolving in a parabola. His opinion indeed was, that all comets revolve in very elongated ellipses; but he remarked, that in any of such cases the orbit near the perihelion does not deviate sensibly from a parabola. By supposing the path of the comet to be parabolic, the investigation of its elements is considerably simplified; but even with this assumption the problem is one of the most difficult in astronomy. It is plain, also, that the parabolic hypothesis cannot assign the major axis of the orbit, nor consequently the time of revolution. Newton, as already stated, remarked that the time of revolution might be found by comparing the intervals which elapsed between the apparitions of comets having the same parabolic elements. For this purpose, it is necessary to form a catalogue of the elements of all those comets the orbits of which have been computed; then, when a new comet has been observed, and its parabolic elements calculated, a reference to the catalogue will serve to indicate whether it has been observed on any former occasion; and if the newly calculated elements should thus turn out to be identical with those of any comet in the catalogue, the interval between the passages of the perihelion will give the time of revolution, supposing the two apparitions to be consecutive. In this way Halley determined the time of revolution of the comet which bears his name; and the same mode of ascertaining the periodicity of a comet is, in consequence of its easy application, constantly practised in the present day. The method most commonly used for computing the parabolic elements of a comet is one invented by the German astronomer Olbers, towards the close of the last century. It is plain that the determination of the elements of a comet's orbit upon the parabolic hypothesis, is subject to the defect of not giving the major axis by direct investigation. In order to ascertain this element, it is necessary that the comet should have been observed at two consecutive passages of the perihelion. Nay, it may happen that, although the comet has been observed on more than one occasion, the apparitions may not be consecutive; and yet there exists no criterion by which a definitive conclusion on this point may be arrived at. Geometers have accordingly investigated methods for computing the elements of a comet's orbit, independently of any hypothesis with respect to the species of conic section in which it may be revolving. In this branch of research, Laplace and Gauss have laboured with eminent success. According to the method devised by the latter geometer, the six elliptic elements of a comet's orbit may be derived from three (in some cases four) observed positions of the body.

When a comet has once been discovered, three observed positions generally suffice, by the aid of Olbers's method, for ascertaining the

F