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ground and invisible; a Membracid (Homoptera) is entirely London, 1897. Many of the memoirs and volumes quoted in the unlike an ant, but is concealed by an ant-like shield. When text also contain further references,

(E. B. P.) we further realize that in this and other examples of

II. CHEMISTRY. Mimicry “the likeness is almost always detailed and remarkable, however it is attained, while the methods The coloration of the surface of animals is caused either differ absolutely,” we recognize that natural selection is by pigments, or by a certain structure of the surface by the only possible explanation hitherto suggested. In the means of which the light falling on it, or reflected through cases of Aggressive Mimicry an animal resembles some its superficial transparent layers, undergoes diffraction or object which is attractive to its prey. Examples are found other optical change. Or it may be the result of a comin the flower-like species of Mantis, which attract the bination of these two causes. It plays an important part insects on which they feed. Such cases are generally in the relationship of the animal to its environment, in described as possessing “alluring colours," and are regarded concealment, in mimicry, and so on; the presence of a

; as examples of Aggressive (Anticryptic) Resemblance, but pigment in the integument may also serve a more direct their logical position is here.

physiological purpose, such as a respiratory function. The Colours displayed in Courtship, Secondary Sexual Char- coloration of birds' feathers, of the skin of many fishes, acters, Epigamic Colours.—Darwin suggested the explana- of many insects, is partially at least due to structure and tion of these appearances in his theory of Sexual Selection the action of the peculiar pigmented cells known as (The Descent of Man, London, 1874). The rivalry of "chromatophores” (which Garstang defines as pigmented the males for the possession of the females he believed cells specialized for the discharge of the chromatic functo be decided by the preference of the latter for those in- tion) and is much better marked when these have for their dividuals with especially bright colours, highly developed background a “reflecting layer" such as is provided by plumes, beautiful song, &c. Wallace does not accept the guanin, a substance closely related to uric acid. Such theory, but believes that natural selection, either directly a mechanism is seen to greatest advantage in fishes. or indirectly, accounts for all the facts. Probably the Among these, guanin may be present in a finely granular majority of naturalists follow Darwin in this respect. The form, causing the light falling on it to be scattered, thus subject is most difficult, and the interpretation of a great producing a white effect ; or it may be present in a proportion of the examples in a high degree uncertain, so peculiar crystalline form, the crystals being known as that a very brief account is here expedient. That selection “iridocytes”; or in a layer of closely apposed needles of some kind has been operative is indicated by the diversity forming a silvery sheet or mirror. In the iris of some of the elements into which the effects can be analysed. fishes the golden red colour is produced by the light The most complete set of observations on Epigamic dis- reflected from such a layer of guanin needles having to play was made by George W. and Elizabeth G. Peckham pass through a thin layer of a reddish pigment, known as upon spiders of the family Attide (Nat. llist. Soc. of a "lipochrome." Again, in some lepidopterous insects a T'i sconsin, vol. i., 1889). These observations afforded the white or a yellow appearance is produced by the deposiauthors"

conclusive evidence that the females pay close tion of uric acid or a nearly allied substance on the surface attention to the love-dances of the males, and also that of the wings. In many animals, but especially among they have not only the power, but the will, to exercise a invertebrates, colouring matters or pigments play an imchoice among the suitors for their favour.” Epigamic portant rôle in surface coloration ; in some cases such characters are often concealed except during courtship; coloration may be of benefit to the animal, but in others they are found almost exclusively in species which are the integument simply serves as an organ for the excretion diurnal or semi-diurnal in their habits, and are excluded of waste pigmentary substances. Pigments (1) may be from those parts of the body which move too rapidly to be of direct physiological importance; (2) they may be seen. They are very commonly directly associated with the excretory; or (3) they may be introduced into the body nervous system; and in certain fish, and probably in other of the animal with the food. animals, an analogous heightening of effect accompanies Of the many pigments which have been described up nervous excitement other than sexual, such as that due to to the present time, very few have been subjected to elefighting or feeding. Although there is Epigamic display mentary chemical analysis, owing to the great difficulties in species with sexes alike, it is usually most marked in attending their isolation. An extremely small amount of those with secondary sexual characters specially developed pigment will give rise to a great amount of coloration, in the male. These are an exception to the rule in heredity, and the pigments are generally accompanied by impurities

, in that their appearance is normally restricted to a single of various kinds which cling to them with great tenacity, sex, although in many of the higher animals they have been so that when one has been thoroughly cleansed, very little proved to be latent in the other, and may appear after of it remains for ultimate analysis. Most of these subthe essential organs of sex have been removed or become stances have been detected by means of the spectroscope, functionless. This is also the case in the Aculeate Hymen- their absorption bands serving for their recognition, but optera when the reproductive organs have been destroyed mere identity of spectrum does not necessarily mean by the parasite (Stylops). Cunningham has recently argued chemical identity, and a few chemical tests have also to be (Sexual Dimorphism in the Animal Kingdom, London, applied before a conclusion can be drawn. The absorp

, 1900) that secondary sexual characters have been produced tion bands are referred to certain definite parts of the by direct stimulation due to contests, &c., in the breeding spectrum, such as the Fraunhofer lines, or they may be period, and have gradually become hereditary, a hypothesis given in wave-lengths. For this purpose the readings of involving the assumption that acquired characters are trans- the spectroscope are reduced to wave-lengths by means of mitted. Wallace suggests that they are in part to be interpolation curves; or if Zeiss's microspectroscope be used, explained as “Recognition Characters,” in part as an in- the position of bands in wave-lengths (denoted by the dication of surplus vital activity in the inale.

Greek letter 1) may be read directly. Examples of the AUTHORITIES.—The following works may also be consulted :- absorption bands yielded by colouring matters will be EIMER. Orthogenesis der Schmetterlinge. Leipzig, 1898.—Poul

found in Ency. Brit. vol. xx. p. 483. The Colours of Animals. London, 1890. —BEDDARD. Animal Coloration.

Hæmoglobin, the red colouring matter of vertebrate London, 1892.-HAASE. Researches on Mimicry (translation). London, 1896. — WALLACE. Naturul

blood, CosH 203 N.S,FeO215, and its derivatives hæmatin, Selection and Tropical Nature. London 1895. Durwinism. C22H%, NFO, and "hæmatoporphyrin, C6H1N,03, are

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colouring matters about which we possess definite chemical it is probably formed by the animal; in other cases it may knowledge, as they have been isolated, purified, and be due to symbiotic algæ, while in the gastric gland of many analysed. Most of the bile pigments of mammals have Mollusca, Crustacea, and Echinodermata, it is derived from likewise been isolated and studied chemically, and all food-chlorophyll. Here it is known as entero-chlorophyll. of these are fully described in the text-books of physio- The black pigments which occur among both vertebrate logy and physiological chemistry. Hæmoglobin, though and invertebrate animals often have only one attribute in physiologically of great importance in the respiratory common, viz., blackness, for among the discordant results process of vertebrate animals, is yet seldom used for of analysis one thing is certain, viz., that the melanins surface pigmentation, except in the face of white races of from vertebrate animals are not identical with those from man or in other parts in monkeys, &c. In some worms invertebrate animals. The melanosis or blackening of the transparent skin allows the hæmoglobin of the blood insect blood, for instance, is due to the oxidation of a to be seen through the integument, and in certain fishes chromogen, the pigment produced being known as a also the hæmoglobin is visible through the integument. uranidine. In some sponges a somewhat similar pigment It is a curious and noteworthy fact that in some inverte- has been noticed. Other pigments have been described, brate animals in which no hæmoglobin occurs, we meet

such actiniochrome, echinochrome, pentacrinin, with its derivatives. Thus hæmatin is found in the antedonin, polyperythrin (which appears to be so-called bile of slugs, snails, the limpet, and the cray- hæmatoporphyrin), the floridines, spongioporphyrin, &c.,

, & fish. In sea-anemones there is a pigment which yields which need no mention here; all these pigments can only some of the decomposition - products of hæmoglobin, be distinguished by means of the spectroscope. and associated with this is a green pigment apparently Most of the pigments are preceded by colourless identical with biliverdin (C16H12N2O4), a green

bile substances known as “chromogens,” which by the action pigment. Again, hæmatoporphyrin is found in the in- of the oxygen of the air and by other agencies become teguments of star-fishes and slugs, and occurs in the changed into the corresponding pigments. In some cases “dorsal streak” of the earth-worm (Lumbricus terrestris), the pigments are built up in the tissues of an animal, in and perhaps in other species. Hæmatoporphyrin and others they appear to be derived more or less directly from biliverdin also occur in the egg-shells of certain birds, but the food. Derivatives of chlorophyll and lipochromes in this case they are derived from hæmoglobin. Hæmo- especially, seem to be taken up from the intestine, probably

. globin is said to be found as low down in the animal by the agency of leucocytes, in which they may occur in kingdom as the Echinoderms, e.g., in Ophiactis virens, combination with, or dissolved by, fatty matters and and Thyonella gemmata. It also occurs in the blood of excreted by the integument. In worms especially, the Planorbis corneus, and in the pharyngeal muscles of skin seems to excrete many effete substances, pigments other mollusca.

included. No direct connexion has been traced between A great number of other pigments have been described ; the chlorophyll eaten with the food and the hæmoglobin for example, in the muscles and tissues of animals, both of blood and muscle. Attention may, however, be drawn vertebrate and invertebrate, are the histohæmatins, of to the work of Dr Schunck, who has shown that a which a special muscle pigment, myohæmatin, is one. In substance closely resembling hematoporphyrin can be vertebrates the latter is generally accompanied by hæmo-prepared from chlorophyll; this is known as phylloporglobin, but in invertebrates—with the exception of the phyrin. Not only does the visible spectrum of this pharyngeal muscles of the mollusca—it occurs alone. substance resemble that of hæmatoporphyrin, but the Although closely related to hæmoglobin or its derivative invisible ultra-violet also, as recently shown by Mr C. A. hæmochromogen, the histohæmatins are yet totally Schunck. distinct, and they are found in animals where not a trace

The reader may refer to Schäfer's Text-Book of Physiology (1898) of hæmoglobin can be detected. Another interesting

for Gamgee's article “On Hæmoglobin, and its Compounds”; to pigment is turacin, which contains about 7 per cent. the writer's papers in the Phil. Trans. and Proc. Roy. Soc. from of nitrogen, found by Professor Church in the feathers 1881 onwards, and also Quart. Journ. Micros. Science and Journ. of of the Cape lory and other plantain-eaters, from which

Physiol.; to Krukenberg's Vergleichende physiologische Studien

from 1879 onwards, and to his Vorträge. Miss Newbigin has it can be extracted by water containing a trace of

collected in Colour in Nature (1898) most of the recent literature ammonia. It has been isolated, purified, and analysed by of this subject. Dr Schuuck's papers will be found under the Professor Church. From it may be obtained turacopor- heading “Contribution to the Chemistry of Chlorophyll” in phyrin, which is identical with hæmatoporphyrin, and gives

Proc. Roy. Soc. from 1885 onwards; and Mr C. A. Schunck's the band in the ultra-violet which Soret and subsequently

paper in Proc. Roy. Soc. vol. lxiii.

(C. A. MACM.) Gamgee have found to be characteristic of hæmoglobin and Columbia, capital of Boone county, Missouri, its compounds. Turacin itself gives a peculiar two-banded U.S.A., situated in 38° 57' N. lat. and 92° 19' W. long., spectrum, and contains about 7 per cent. of copper in its in the central part of the state, on the Wabash Railway, molecule. Another copper-containing pigment is hæmo- at an altitude of 783 feet. It is the site of the State cyanin, which in the oxidized state gives a blue colour to University, and of Christian and Stephens Female Colleges. the blood of various mollusca and arthropoda. Like Population (1880), 3326 ; (1890), 4000; (1900), 5651, . hæmoglobin, it acts as an oxygen-carrier in respiration, but it takes no part in surface coloration.

Columbia, a borough of Lancaster county, PennA class of pigments widely distributed among plants sylvania, U.S.A., situated on the east bank of Susquehanna and animals are the lipochromes. As their name denotes,

river, in the south-eastern part of the state, on branches they are allied to fat and generally accompany it, being Railways, at an altitude of 251 feet.

of the Pennsylvania and the Philadelphia and Reading soluble in fat solvents, They play an important part manufactures, principally of iron.

It has extensive in surface coloration, and may be greenish, yellow, or red

Population (1880), in colour. They contain no nitrogen. As an example of

8312; (1890), 10,599; (1900), 12,316. a lipochrome which has been isolated, crystallized, and Columbia, capital of Richland county, South purified, we may mention carotin, which has recently been Carolina, U.S.A., and of the state, situated in 34° 00' found in green leaves. Chlorophyll

, which is so often N. lat. and 80° 57' W. long. on the east bank of associated with a lipochrome, has been found in some Congaree river, at the junction of the Saluda and Broad, Infusoria, and in Hydra and Spongilla, &c. In some cases near the centre of the state, at an altitude of 244 feet.

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COMBINATORIAL ANALYSIS Five railways enter it, namely, the Atlantic Coast Line, Columbus, capital of Lowndes county, Mississippi, the Southern, the South Carolina and Georgia, the Florida U.S.A., at the intersection of the Southern and the Mobile Central and Peninsula, and the Columbia, Newberry, and and Ohio Railways, on the Tombigbee river. It contains Laurens. It is the seat of South Carolina College, which large cotton mills. Population (1890), 4559; (1900, with in 1898 had twelve professors and 188 students. Popula- limits enlarged), 6484, including 3366 negroes. tion (1880), 10,036; (1890), 15353 ; (1900), 21,108.

Columbus, capital of Franklin county, Ohio, Columbia, capital of Maury county, Tennes

U.S.A., and of the state of Ohio. The site was pursce, U.S.A., situated on Duck river, in the central posely selected in 1813 near the centre of the state,

in part of the state, at an altitude of 646 feet. It has two

39° 57' N. lat. and 82° 59' W. long., at an railways, the Louisville and Nashville and the Nashville, altitude of 743 feet, the elevation of the lines at the Union Chattanooga, and St Louis. Population (1880), 3400;

Station. It is a railway centre of the first importance. (1890), 5370; (1900), 6052.

Fourteen different lines of railway, belonging to eight

companies, enter the fine new Union Station in the heart Columbia, District of. See WASHINGTON. of the city, thence radiating in all directions.

The manuColumbia University, in the city of New York,

factures employed in 1890 a capital of $16,178,703, with U.S.A., includes both a college and a university in the strict

13,421 hands and an output of $22,887,586. The prinsense of the word as used in the United States. It com

cipal manufacture was that of carriages and waggons, valued the faculties of law, medicine, philosophy, political

at $3,199,287. Foundry and machine-shop products were science, pure science, and applied science. It is the successor

second, with a value of $2,139,185. Then followed the of the corporation known as “The Governors of King's

manufacture of steam cars, the product of which was valued College, in the Province of New York,” founded in 1754 by

at $1,670,078. The Ohio State University, situated here, royal charter. In the educational system is also included

had in 1898 a faculty of 95 professors, and was attended Barnard College for Women, a separate corporation founded

by 1150 students, one-fifth of whom were women. Its in 1889, and a teachers' college, also a separate corpora

property was valued at $2,600,000, and its income at tion. In 1897 the university moved from the centre of

$292,000. It has schools of law, medicine, dentistry, the city northwards to Morningside Heights, which over

pharmacy, and veterinary surgery. Capital L'niversity, a look feet. The

Lutheran institution, also here, had in 1898 a faculty of )

10 and 113 . The death-rate in remains in its old location opposite Roosevelt Hospital. The entire plant of the university represents a cost of

average of American cities, and little more than half that about $9,500,000. In the year ending 30th June 1900

of the Union. The assessed valuation of property, real the expenditures for educational purposes were $942,460,

and personal, was, in 1899, $64,344,990. The income from leaving a deficiency of $17,328, which was met by a

all sources was $2,612,301, the expenditure $2,570,038, special guarantee fund. In 1901 there were registered in

and the net debt $6,059,146. The tax rate per $1000 the college for men (Columbia) 475 students, and in the

was $27.50. Population (1880), 51,647; (1890), 88,150; college for women (Barnard) 293 students, making a total

(1900), 125,560; death-rate (1900), 15.8. of 768 undergraduates. The total of non-professional Combaconum, or Kumbakonam, a city of graduate students was 412. The scientific schools con- British India, in the Tanjore district of Madras, in the tained 539 students, the law school 422 students, the delta of the Kaveri ; with a railway station on the South medical school 775 students, the teachers' college 498 Indian Railway, 194 miles from Madras. In 1881 it had students—making a total of 2234 professional students. a population of 50,098, and in 1891 of 54,307, of whom The total number of students in the university was thus nearly one-fifth were Brahmans. In 1901 the population 3830 (including 417 summer session students). In addition was 59,688, showing an increase of 10 per cent. The to these there were 29 auditors and 679 members of exten- municipal income in 1897–98 was Rs.80,480. It contains sion courses, making a grand total of 4538. The number a Government college, two high schools, four printingof teachers of all grades for the same year was 375. The presses, and a reading-room. library, which numbers about 300,000 volumes, is thoroughly Combinatorial Analysis.- The Combinamodern, and is selected with special reference to scholarly

torial Analysis, as it was understood up to the end of the uses. The university is growing in all departments. (See

18th century, was of limited scope and restricted also UNIVERSITIES and EDUCATION.)

(s. L*)

application. P. Nicholson, in his Essays on the Historical

Combinatorial Analysis, published in 1818, duction. Columbus, capital of Muscogee county, Georgia, U.S.A., situated on the western boundary of the state, at

states that “the Combinatorial Analysis is a an altitude of 260 feet, on Chattahoochee river, which is

branch of mathematics which teaches us to ascertain and navigable to this point. Just above the city the river

exhibit all the possible ways in which a given number crosses the fall line, producing falls and rapids which

of things may be associated and mixed together; so that furnish excellent water-power. This has been turned to

we may be certain that we have not missed any collection account in extensive cotton manufactures. Three railways

or arrangement of these things that has not been enumer

ated.” Writers on the subject seemed to recognize fully enter the city, the Southern, the Central of Georgia, and the Georgia and Alabama. Population (1880), 10,123 ;

that it was in need of cultivation, that it was of much (1890), 17,303 ; (1900), 17,614.

service in facilitating algebraical operations of all kinds,

and that it was the fundamental method of investigation Columbus, capital of Bartholomew county, Indiana, in the theory of Probabilities. Some idea of its scope U.S.A., situated on the cast fork of White river, a little may be gathered from a statement of the parts of algebra south of the centre of the state, at an altitude of 629 feet. to which it was commonly applied, viz., the expansion of It is at the intersection of lines of the Pittsburg, Cincinnati, a multinomial, the product of two or more multinomials, Chicago, and St Louis and the Cleveland, Cincinnati, the quotient of one multinomial by another, the reversion Chicago, and St Louis Railways. The centre of population and conversion of series, the theory of indeterminate of the United States was in 1900 very near this place. equations, &c. Some of the elementary theorems and Population (1880), 4813; (1890), 6719; (1900), 8130. various particular problems appear in the works of the

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earliest algebraists, but the true pioneer of modern partition of numbers. Other branches of combinatorial researches seems to have been Abraham Demoivre, who analysis were, from any general point of view, absolutely first published in 1697 (Phil. Trans. R. S.) the law of neglected. In 1888 MacMahon investigated the general the general coefficient in the expansion of the series problem of distribution, of which the partition of a number a + bx + cx? + dx3 + ... raised to any power. (See also is a particular case. He introduced the method of symMiscellanea Analytica, Bk. iv. chap. ii. prob. iv.) His metric functions and the method of differential operators, work on Probabilities would naturally lead him to consider applying both methods to the two important subdivisions, questions of this nature. An important work at the tim

An important work at the time the theory of composition and the theory of partition. it was published was the De Partitione Numerorum of He introduced the notion of the separation of a partiEuler, in which the consideration of the reciprocal of tion, and extended all the results so as to include multithe product (1 – x2)(1 – x-2)(1 – 232)... establishes a funda- partite as well as unipartite numbers. He showed how mental connexion between arithmetic and algebra, arith- to introduce zero and negative numbers, unipartite and metical addition being made to depend upon algebraical multipartite, into the general theory; he extended multiplication, and a close bond is secured between the Sylvester's graphical method to three dimensions; and theories of discontinuous and continuous quantities. The finally, 1898, he invented the "Partition Analysis” and multiplication of the two powers 29, xb, viz., 29 x ab = x2 +b applied it to the solution of novel questions in arithmetic

х showed Euler that he could convert arithmetical addition and algebra. An important paper by G. B. Mathews, into algebraical multiplication, and in the paper referred which reduces the problem of compound partition to that to he gives the complete formal solution of the main of simple partition, should also be noticed. This is the problems of the partition of numbers. He did not obtain problem which was known to Euler and his contemporaries general expressions for the coefficients which arose in the as “the Problem of the Virgins," or the Rule of Ceres”; expansion of his generating functions, but he gave the it is only now, nearly 200 years later, that it has been actual values to a high order of the coefficients which solved. arise from the generating functions corresponding to various The most important problem of combinatorial analysis conditions of partitionment. Other writers who have is connected with the distribution of objects into classes. contributed to the solution of special problems are James A number n may be regarded as enumerating n Bernouilli, Boscovitch, Hindenburgh, Emerson, Wood- similar objects; it is then said to be unipartite. Fundahouse, Simpson, and Barlow. Problems of combination On the other hand, if the objects be not all were generally undertaken as they became necessary for similar they cannot be effectively enumerated the advancement of some particular part of mathematical | by a single integer; we require a succession of integers. science : it was not recognized that the theory of combina- If the objects be p in number of one kind, 9 of a second tions is in reality a science by itself, well worth studying kind, r of a third, &c., the enumeration is given by the for its own sake irrespective of applications to other parts succession par ... which is termed a multipartite number, of analysis. There was a total absence of orderly develop- and written ment, and until the first third of the 19th century had

Par passed, Euler's classical paper remained alike the chief

where p +9+r+

If the order of magnitude result and the only scientific method of combinatorial

of the numbers P, 9, 9, analysis.

is immaterial, it is usual to In 1846 Jacobi studied the partitions of numbers by write them in descending order of magnitude, and the means of certain identities involving infinite series that are

succession may then be termed a partition of the number met with in the theory of Elliptic Functions. The method

n, and is written (par...). The succession of integers employed is essentially that of Euler. Interest in England thus has a twofold signification : (i.) as a multipartite was aroused, in the first instance, by De Morgan in 1846, number it may enumerate objects of different kinds; (ii.) who, in a letter to Henry Warburton, suggested that it may be viewed as a partitionment into separate parts of combinatorial analysis stood in great need of development, a unipartite number. We may say either that the objects

, and alluded to the theory of partitions. Warburton, to

are represented by the multipartite number par ..., or that some extent under the guidance of De Morgan, prosecuted they are defined by the partition (par... ) of the unipartite researches by the aid of a new instrument, viz., the theory

number n. Similarly the classes into which they are disof finite differences. This was a distinct advance, and he tributed may be m in number all similar; or they may was able to obtain expressions for the coefficients in parti-be P1 of one kind, 11 of a second, r, of a third, &c., tion series in some of the simplest cases (Trans. Camb. where P1+11+r+. We may thus denote the Phil. Soc., 1849). This paper inspired a valuable paper classes either by the multipartite numbers P101"1 ..., or by by Sir John Herschel (Phil. Trans. Roy. Soc., 1850), who, the partition (P12191 ... ) of the unipartite number m. The by introducing the idea and notation of the circulating distributions to be considered are such that any number of function, was able to present results in advance of those objects may be in any one class subject to the restriction of Warburton. The new idea involved a calculus of the that no class is empty. Two cases arise. If the order imaginary roots of unity. Shortly afterwards, in 1855, of the objects in a particular class is immaterial the class the subject was attacked simultaneously by Cayley and is termed a parcel ; if the order is material the class is Sylvester, and their combined efforts resulted in the termed a group. The distribution into parcels is alone practical solution of the problem that we have to-day. considered here, and the main problem is the enumeration The former added the idea of the prime circulator, and of the distributions of objects defined by the partition the latter applied Cauchy's theory of Residues to the (par...) of the number n into parcels defined by the subject, and invented the arithmetical entity termed a partition (2,9191 ... ) of the number m. (See “Symmetric “denumerant.” The next distinct advance was made by l'unctions and the Theory of Distributions,” Proc. London Sylvester, Franklin, Durfee, and others, about the year Mathematical Society, vol. xix.) Three particular cases 1882 (Amer. Journ. Math. vol. v.) by the employment of are of great importance. Case I. is the one-to-one dis

“ a graphical method. The results obtained were not only tribution,” in which the number of parcels is equal to the valuable in themselves, but also threw considerable light number of objects, and one object is distributed in each upon the theory of algebraic series. So far it will be seen parcel. Case II. is that in which the parcels are all that researches had for their object the discussion of the different, being defined by the partition (1111 ... ), con

S. III.

= n.

...=m.

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...

veniently written (1m); this is the theory of the composi- change of object and parcel we arrive at the well-known theorem tions of unipartite and multipartite numbers. Case III. of symmetry in symmetric functions, which states that the cois that in which the parcels are all similar, being defined product anyag,arı... in a series of monomial symmetric functions,

efficient of symmetric function (pqr ...) in the development of the by the partition (m); this is the theory of the partitions of is equal to the coefficient of the function (P19.71 ...) in the similar unipartite and multipartite numbers. Previous to dis- development of the product apaqar.... cussing these in detail, it is necessary to describe the

The general result of Case I. may be further analysed with method of symmetric functions which will be largely

important consequences. utilized.

Write

X=(1)21, Let a, B, Y,... be the roots of the equation

X,=(2.9+(1),

X3=(3)āz+(21)x X + (13) = 0.

and generally The Distri- The symmetric function LaB'y"

Xs=3(Xuv....amma.com

where p+9+9+ =n is

, in the partition notation, the summation being in regard to every partition of s. Consider Function.

denote written (par ...). Let A,

the result of the multiplication

(par...), (P19171...) the number of ways of distributing the n objects defined by

Xni42.871. = Pæ the partition (Por...) into the m parcels defined by the

To determine the nature of the symmetric function P a few partition (P1971 ...). The expression

definitions are necessary.
ΣΑ,
(Pqr ...),

Definition 1.Of a number n take any partition (9,121 z ... /s) (par...), (P1917ı...)

and separate it into component partitions thus :where the numbers P1, 91, 91 are fixed and assumed to

(λλ.) (λληλ:)(λο) ... be in descending order of magnitude, the summation being in any manner. This

may be termed a separation of the partition, for every partition (Par ... ) of the number n, is defined to the numbers occurring in the separation being identical with be the distribution function of the objects defined by those which occur in the partition. In the theory of symmetric (par...) into the parcels defined by (2,27,...). It gives a

functions the separation denotes the product of symmetric

functionscomplete enumeration of n objects of whatever species into parcels of the given species.

{a^ip123a^38ddy 155a16 ... 1. One-to-One Distribution. Parcels m in number (2.e. m

m=n).
The portions (9/12), (131x\5), (16), are termed separates, and if

be in descending order of
Let ho be the homogeneous product-sum of degree's of 21+13=21, 13+14+ 1;=11, 16=rı,
the quantities a, ß, 7, .

magnitude, the usual arrangement, the separation is said to have so that

a species denoted by the partition (P191?'1 ...) of the number 1. (1 - ax . 1- Bx. 1-yx....)-1=1+ha+hxx2 + 11223 +...

Definition 11.if in any distribution of objects into n parcels h, = Σα = (1)

(one object in each parcel), we write down a number ğ, whenever h, =Σα2 + Σαβ= (2) + (12)

we observe ç similar objects in similar parcels we will obtain a lt = Σα3 + Σαββ+ Σαβγ= (3)+ (21)+ (13).

succession of numbers ši, 82, 83, .,,, where ($1$2$3 ...) is some parti

tion of n. The distribution is then said to have a specification Form the product hpıhahri...

denoted by the partition ($1$283 ...); Any term in hpı may be regarded as derived from P, objects dis- Now it is clear that P consists of an aggregate of terms, each of tributed into P. similar parcels, one object in each parcel, since which, to a numerical factor près, is a separation of the partition the order of occurrence of the letters a, b, Y, ... in any term is immaterial. Moreover

, every selection of a letters from the ($97.*. ...) of species (2191", ...). Further, P is the distribution letters in apply? ... will occur in some term of hpı, every further function of objects into parcels denoted by (P19191 ...), subject to selection of a letters will occur in some term of hai, and so on.

the restriction that the distributions have each of them the Therefore in the product hpinaihri the term apply" and there- specification denoted by the partition (s ...). Employing fore also the symmetric function (pqr ...) will occur as many times as it is possible to distribute objects defined by (pqr ...) into a more general notation we may write parcels defined by (P19191 ...) one object in each parcel. Hence

X"X">XT'.

= SP Apr...), (un...). (par ... )=hphalerı ...

and then P is the distribution function of objects into parcels This theorem is of algebraic importance ; for consider the simple particular case of the distribution of

objects (43) into parcels

(52), (2,720 * 2.2...), the distributions being such as have the speciand represent objects and parcels by small and capital letters respectively. One distribution is shown by the scheme

fication ($97.06.202...). Multiplying out P so as to exhibit it as a A A A A ABB

sum of monomials, we get a result, а а а а и wherein an object denoted by a small letter is placed in a parcel

X"2" X3...=

= 230(9???????...). denoted by the capital letter immediately above it. We may interchange small and capital letters and derive from it a distribu- indicating that for distributions of specification (smas ..) tion of objects (52) into parcels (13); viz. :

there arc 0 ways of distributing n objects denoted by (

akt...) AAAABBB ααα α α οι"

amongst n parcels denoted by (

225...), ono object in each

n 22.2 The process is clearly of general application, and establishes a oneto-one correspondence between the distributions of objects (Pqr ... ) and object, and that this operation leaves the specification of the

parcel. Now observe that as before we may interchange parcel into parcels (029171 ...) and the distributions of objects (P1!11 ...) | distributions unchanged. Hence the number of distributions into parcels (par ...). It is in fact, in Case I., an intuitive observa

must be the same, and if tion that we may either consider an object placed in or attached to a parcel, or a parcel placed in or attached to an object. Analyti

X","?X3

+0(1) cally we have

then also Theorem.—“The coefficient of symmetric function (par ...) in the development of the product hpın qıhtı... is equal to the coefli

.)

+... cient of symmetric function (P19ırı ...) in the development of the

19 13 product hphqhr

This extensive theorem of algebraic reciprocity includes many The problem of Case I. may be considered when the distribu- known theorems of symmetry in the theory of Symmetric tions are subject to various restrictions. If the restriction be to Functions. the effect that an aggregate of similar parcels is not to contain The whole of the theory has been extended to include synmore than one object of a kind, we have clearly to deal with the

metric functions symbolized by partitions which contain as well clementary symmetric functions an, Q2, Q.3, or (1), (12), (13), ... zero and negative parts. in lieu of the quantities N1, h2, h3, The distribution function

2. The Compositions of Multipartite Numbers. Parcels denoted by Las then the value apaq ari.

... or (1P1) (191) (1'?)..., and by inter-(1m).—There are here no similarities between the parcels.

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