304 L A P L A P
than r. It can be easily proved that T satisfies Laplace's differential equation –
f f-T <? 2 (rT) -r-r, + r j-y- = ;
and if for T we substitute the expanded form, we obtain the general differential equation of which Laplace's coefficients are particular integrals: –
dp. 1 - /it - d(a i i
Expressions which satisfy this equation[1] are referred to as Laplace's functions; they include as a particular case the coefficients, which are, as we have seen, certain definite functions of the spherical surface coordinates of the two points. If
X = fifjf + Vl - /t 2 Vl - fJ. 2 COS (w - ft/),
the coefficients become functions of x alone, and it was in this form that Legendre first introduced them. One of the fundamental properties of Laplace's functions, known as Laplace's theorem, is that, if Y; and Z/ be two such functions, i and i being whole numbers and not identical, then
iZi duda> = .
Again, if Y/ is the same function of // and w , that Yi is of jit and w, we have the important relation
But the, property on which their utility in physical researches chiefly depends is that every function of the coordinates of a point on a sphere can be expanded in a series of Laplace's functions.[2]
In the figure of the earth, the theory of attractions, and the sciences of electricity and magnetism this powerful calculus occupies a prominent place. Gauss in particular has employed it in the calculation of the magnetic potential of the earth, and it has recently received new light from Professor Clerk Maxwell's interpretation of harmonics with reference to poles on the sphere.
Laplace, always profound rather than elegant, nowhere displays the massiveness of his genius so conspicuously as in the theory of probabilities. The science which Pascal and Fermat had initiated he brought very nearly to perfection; but the demonstrations are so involved, and the omissions in the chain of reasoning so frequent, that the Théorie Analytique is to the best mathematicians a work requiring the most arduous study. The theory of probabilities, which Laplace describes as common sense expressed in mathematical language, first attracted his attention from its importance in physics and astronomy; and he applies his theory, not only to the ordinary problems of chances, but also to the inquiry into the causes of phenomena, vital statistics, and future events.
The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
The method of generating functions, the foundation of his theory of probabilities, Laplace published in 1779; and the first part of his Théorie Analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable. The latter is therefore called the generating function of the former. A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients. The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences. The method, however, is now obsolete from the more extended facilities afforded by the calculus of operations.
The first formal proof of Lagrange's theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality. He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved. He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.
In 1842, the works of Laplace being nearly out of print, his widow was about to sell a farm in order to procure funds for a new impression, when the Government of Louis Philippe took the matter in hand. A grant of 40,000 francs having been obtained from the chamber, a national edition was issued in seven 4to vols., bearing the title Œuvres de Laplace, 1843-47. The Mécanique Céleste with its four supplements occupies the first 5 vols., the 6th contains the Système du Monde, and the 7th the Th. des Probabilités, to which the more popular Essai Philosophique forms an introduction. Of the four supplements added by the author, 1816-25, he tells us that the problems in the last were contributed by his son. An enumeration of Laplace's memoirs and papers (about one hundred in number) is rendered superfluous by their embodiment in his principal works. The Th. des Prob. was first published in 1812, the Essai in 1814; and both works as well as the Système du Monde went through repeated editions. Laplace's first separate work, Théorie du Mouvement et de la Figure Elliptique des Planètes, 1784, was published at the expense of President Bochard de Sarou. The Précis de l'Histoire de l'Astronomie, 1821, formed the fifth book of the 5th edition of the Système du Monde. An English translation, with copious elucidatory notes, of the first 4 vols. of the Mécanique Céleste, by Dr Bowditch, was published at Boston, U.S., 1829-39, in 4 vols. 4to; a compendium of certain portions of the same work by Mrs Somerville appeared in 1831, and a German version of the first 2 vols. by Burckhardt at Berlin in 1801. English translations of the Système du Monde by Mr Pond and Mr Harte were published, the first in 1809, the second in 1830. An edition entitled Les Œuvres Complètes de Laplace, 1878, &c., which is to include the whole of his memoirs, is now in course of publication under the auspices of the Academy of Sciences. The four 4to vols. which have already appeared comprise the first ten books of the Mécanique Céleste.
Scanty notices of Laplace's life will be found in Fourier's Éloge, June 15, 1829, in the funeral oration of Poisson, and Arago's Report, 1842, translated amongst his Biographies by Admiral Smyth and Mr Grant. His astronomical work is treated of in Gautier's Problème des trois Corps and Grant's Hist. of Astronomy. For Laplace's functions see Dr E. Heine, Handbuch der Kugelfunctionen, Berlin, 1861; John H. Pratt, A Treatise on Attractions, 1865; Todhunter's Elementary Treatise on Laplace's Functions, 1875, and History of the Mathematical Theories of Attraction, 1873; N. M. Ferrers's Elementary Treatise on Spherical Harmonics, 1877; and L. Schläfli, Die zwei Heine'schen Kugelfunctionen, 1881. Consult also Thomson and Tait, Treatise on Natural Philosophy, 1879, p. 141; Clerk Maxwell, Treatise on Electricity, chap. ix; Professor Niven in Phil. Trans., 1879, p. 379; Dirichlet in Crelle, xvi. p. 35; and Jacobi, vol. ii. p. 223, xxvi. p. 82. Some of Laplace's results in the theory of probabilities are simplified in Lacroix's Traité élémentaire du Calcul des Probabilités and De Morgan's Essay, published in Lardner's Cabinet Cyclopædia. For the history of the subject see A History of the Mathematical Theory of Probability, by Isaac Todhunter, 1865. (A. M. C.)
LAPLAND, or Lappland, is the north-west portion of the continent of Europe, bounded W., N., and E. by the North Atlantic, the Arctic Ocean, and the White Sea, and S. partly by the White Sea, but mainly by a conventional line. It includes the northern parts of Norway, Sweden, and Finland, and the western part of the Russian government of Archangel. A line drawn from the mouth of the Salten Fjord on the Norwegian coast to the mouth of the Ponoi on the White Sea, practically identical with the 61st parallel of north latitude, measures 700 miles. Of Russian Lapland only a very small portion lies outside of the Arctic circle; but in Swedish Lapland the southern confines descend as low as 64°. According to Frijs (in Petermann's Mitth., 1870), the total area of Lapland may be estimated at 153,200 square miles, of which 16,073 miles belong to Norway, 48,898 to Sweden, 26,575 to Finland, and 61,654 to Russia.
Lapland is merely the land of the Lapps or Laps, and does
- ↑ This equation was first integrated by Mr Hargreave, Philosophical Transactions, 1841, p. 75, and it has since been successfully treated by Professor Boole, Camb. and Dub. Math. Journ.,mm vol. i. p. 10, and Professor Donkin, Phil. Trans., 1857, p. 43. See Boole's Differential Equations, 3d ed., p. 433.
- ↑ The proof of this theorem in its full generality has given rise to much controversy; where, however, the form of the function is rational and integral – the only case of practical importance – no difficulty is experienced. The reader is referred to two papers by Ivory in the Phil. Trans., 1812 and 1822; Poisson, Théorie Mathématique de la Chaleur; L. Dirichlet, in Crelle's Journal, vol. xvii.; and O. Bonnet, in Liouville's Journal, vol. xvii.
