01 



COMBINING PKOPORTION. 



COMETS. 





letters, allowing any letter to be repeated in all or or any of the places, 

 is a?, or the product of x, x, x, . . . . ( factors in all). This is the 

 number of permutations of out of .>, allowing repetition. 



VII. The number of ways in which n different letters can be distri- 

 buted into x boxes, all possible modes of distribution being equally 

 allowable, is a?. 



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 2* 1, or 15 different sections : they are 



abed, bed, acd, abd, a be, ab, ac, ad, be, Id, cd, 

 a, b, c, d. 



Among the curiosities of this subject, it will suffice to mention tha 

 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 per- 

 mit possible alterations of spelling, M. de Hainan computed that the 

 word llainaut might be spelt in 2304 different ways, so as to be 

 pronounced in the same way by as many different Frenchmen, or very 

 nearly so. 



The most useful proposition in the higher part of the theory of 



combinations is the reduction of the formula 1. 2. 3 (x 1) x to 



a very close approximation, which can be easily calculated by logarithms. 

 It affords at the same time a useful lesson to those who have not 

 studied mathematics at all, or very little ; we have seen ignorance 

 comport itselJ with laughter more than once at the idea of the pre- 

 ceding product being found by employing the proportion which the 

 circumference of a circle bears to its diameter. But let * = 3'141593 

 be this proportion ; = 2 "718282, the base of Napier's logarithms : 

 then we have 



1.2.3 (x 1) x= V2 x f-^-j very nearly. 



which is a little too small, but the error is only about the 12j-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 loga- 

 rithms of this product will be found at the end of the article ' Theory 

 of Probabilities,' in the ' Encyclopedia Metropolitana.' For an instance 

 of the computation, see the Library of Useful Knowledge : ' Examples 

 of Arithmetic,' &c., p. 45. 



COMBINING PROPORTION. [ATOMIC THEORY.] 



( < iMBINING VOLUME. [ATOMIC VOLUMK.] 



< ( >.\I Ul.'STIBLE. 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. 



[COMBUSTIOX.] 



COMBUSTION is a term usually restricted to describe the pheno- 

 menon that ensues when chemical action is sufficiently intense to pro- 

 ilix. light and heat. The burning of coal, wood, paper, candles, oil, 

 l-gas are familiar illustrations of combustion. Less common, but 

 i illi, uit, instances of combustion are seen in the explosion of 

 gunpowder, or fireworks, or in the burning of steel-wire, charcoal, or 

 pbcnphonu in oxygen gas. 



In the examples of combustion above alluded to, the action lies 

 Ix-twi-en the burning body on the one hand, and pure or diluted oxygen 

 mi the other; and inasmuch as our world b enveloped in an atmosphere 

 n[ which the most important constituent is oxygen, it follows that all 

 ordinary instances of combustion arc owing to the rapid oxidisation 

 of bodies at a high temperature. It would be wrong, however, to 

 -e that the word combustion expresses no other actions than 

 indicated. Many substances burn equally well in atmospheres 

 frrn which oxygen is excluded altogether, and in some cases even 

 burn more readily than they would under similar circumstances in 

 ]i':i- oxygen. For instance, when the metals arsenic or antimony are 

 finely powdered and thrown into an atmosphere of chlorine, they 

 instantly ignitp, and burn with evolution of light and heat ; in fact, 

 literally undergo combustion. 



Combuttililes, and tnppartert of mmbutlion or non-combititllles, 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 con- 

 ventional, and nly applicable so long as the circumstances under 

 which it i 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 com- 

 biutinn will then go on at the jet : the air may now with equal pro- 

 prietor be said to be the combustible and the gas the non-combustible, 

 for tRe 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 



combustion of each other. In like manner a jet of mercury vapour la 

 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 phos- 

 phorus, 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 accumu- 

 lation of materials of that description. 



The spontaneous combustion of the human body is a subject that 

 properly belongs to the past century ; nevertheless, as some ignorant 

 people at the present day believe that excessive and long-continued 

 drinking of ardent spirits sets up a condition of system under which 

 spontaneous burning of the body may ensue, it may be as well to notice 

 here that, before the causes of combustion were investigated, it was 

 customary when any person was found burnt to death and the origin 

 of the fire could not be discovered, to assume that combustion had 

 occurred spontaneously. The hypothesis has, however, long since been 

 found to be untenable, and, amongst intelligent persons, is now only 

 held by a few who have not taken the trouble to acquaint themselves 

 either with the laws of heat or the causes of the phenomenon of 

 burning. 



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, how- 

 ever, 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.] 



ciiMENIC ACID. [MECONIC Aero.] 



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 last- 

 mentioned appendage, is denominated the head. The outline of the 

 head is hazy and ill defined ; hence the origin of the term comet 

 ((to^TTjt, from K/>tn\, 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 pro- 

 longation 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 Brahe', is said to have exhibited neither tail nor coma, but 

 appeared perfectly round like a planet. (" Plane rotunda extitit ; nee 

 ullam caudam aut barbam in unam magis quarn in aliam partem 

 portendebat." *) Cassini relates a similar fact with respect to the 

 coineta 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. 



