Page:EB1911 - Volume 27.djvu/992

 In the theory of attractions this expression is interpreted as measuring the degree of attenuation of the quantity at P; if we reverse the sign we get the concentration, −∇2.

Again, if we form the scalar product of the operator V into a vector A we have

∇A＝ -1-j'-gl;-l-ki) =~'?;§ %+5°, ,4'§ +%. . (20)

If A represent the velocity at any point (x, y, z) of a fluid, the latter expression measures the rate at which fluid is flowing away from the neighbourhood of P. By a generalization of this idea, it is called the divergence of A, and we write

The vector product [VA] has also an important significance. We find

vm = [ 0%-I-ji, -I-kg; 

6A3 BA;, GA, BA; GA2 BA,

"I ély 02) +-l<6z  dx) +k<6x By) ' (22)

If A represent as before the velocity of a fluid, the vector last written will represent the (doubled) angular velocity of a fluid element. Again if A represent the magnetic force at any point of an electro-magnetic field, the vector [V ] will represent the electric current. In the general case it is called the curl, or the rotation, of A, and we write

vA]=curl A, or rot A. .. . (23)

These definitions enable us to give a compact form to two important theorems of C. F. Gauss and Sir G. G. Stokes. The former of these may be written,

div A. dV=fAndS,. . . (24)

where the integration on the left hand includes all the volume elements dV of a given region, and that on the right includes all the surface-elements dS of the boundary, n denoting a unit vector drawn outwards normal to dS. Again, Stokes's theorem takes the form /'Ads=f curl A ndS,. . . (25)

where the integral on the right extends over any open surface, whilst on the left ds is an element of the boundin curve, treated as a vector. A certain convention is implied as to tliie relation between the positive directions of n and ds.

It is to be observed that the term “vector ” has been used to include two distinct classes of geometrical and physical entities. The first class is typified by a displacement, or a mechanical force. A polar vector, as it is called, is a magnitude associated with a certain linear direction. This may be specified by any one of a whole assemblage of parallel lines, but the two “senses” belonging to any one of the lines are distinguished. The members of the second class, that of axial vectors, are primarily not vectors at all. An axial vector is exemplified by a couple in statics; it is a magnitude associated with a closed contour lying in any one of a system of parallel planes, but the two senses in which the contour may be described are distinguished. It was therefore termed by H. Grassmann a Plangrosse or Ebenengrosse. ]ust as a polar vector may be indicated by a length, regard being paid to its sense, so an axial vector may be denoted by a certain area, regard being paid to direction round the contour. A theory of “ Plangrossen ” might be developed throughout on independent lines; but since the laws of combination prove to be analogous to those of suitable vectors drawn perpendicular to the respective areas, it is convenient for mathematical purposes to include them in the same calculus with polar vectors. In the case of couples this procedure has been familiar since the time of L. Poinsot (1804). In the Cartesian treatment of the subject no distinction between polar and axial vectors is necessary so long as we deal with congruent systems of co-ordinate axes. But when we pass from a right-handed to a left-handed system the formulae of transformation are different in the two cases. A polar vector (e.g. a displacement) is reversed by the process of reflection in a mirror normal to its direction, whilst the corresponding axial vector (e. a couple) is unaltered.

.-The methods of vector analysis are chiefly used as a means of condensed expression of various important relations which are of frequent occurrence in mathematical physics, more especially in electricity. They are freely employed, for example, in many recent German treatises. The historical development of the subject can only be briefiy referred to. The notions of scalar and vector products originated independently with Sir W. R. Hamilton (1843) (see ) and H. Grassmann (1844), but were associated with various other conceptions of which no use is made in the simplified system above sketched. The present currency of this latter Zystem is due mainly to the advocacy of O. Heaviside and ]. W. ibbs, although for the systematic hysical interpretation of the various combinations of symbols which constantly recur in electricity and allied subjects we are indebted primarily to the classical treatise of J. C. Maxwell on Electricity and Magnetism (1873). For further details and applications of the calculus reference may be made to the following: O. Heaviside, Electro-Magnetic Theory (London, 1894); J. W. Gibbs, Vector Analysis (2nd ed., New York, 1907); M. Abraham, Die Maxwellsche Theorie d. Elektrizität (Leipzig, 1904); the articles by H. E. Timerding and M. Abraham in vol. iv. of the ''Encycl. d. Math. Wiss.'' (Leipzig, 1901–2); A. H. Bucherer, Elemente d. Vektor-Analysis (Leipzig, 1905). For an account of other systems of vector analysis see H. Hankel, Theorie d. complex en Zahlensysteme (Leipzig, 1867); and A. N. Whitehead, Universal Algebra, vol. i. (Cambridge, 1898).

VEDDAHS, or (from Sanskrit veddha, “hunter”), a primitive people of Ceylon, probably representing the Yakkos or “demons” of Sanskrit writers, the true aborigines of the island. During the Dutch occupation (1644–1796) they were found as far north as Jaffna, but are now confined to the south-eastern district, about the wooded Bintenna, Badulla and Nilgala hills, and thence to the coast near Batticaloa. They are divided into two classes, the Kele Weddo or jungle Veddahs, and the Gan Weddo, or semi-civilized village Veddahs. The Veddahs exhibit the phenomenon of a race living the wildest of savage lives and yet speaking an Aryan dialect. Craniometrical evidence strongly favours the theory, now generally accepted, that they represent a branch of the pre-Aryan Dravidians of southern India, and that their ancestors probably made a settlement in the island of Ceylon in prehistoric times, detaching themselves from a migrating horde which passed through the island to find at last a permanent home in the continent of Australia. The true jungle veddahs are almost a dwarfish race. They are dark-skinned and flat-nosed, slight of frame and very small of skull, and average no more than 5 ft. Their black hair is shaggy rather than lank. They are a shy, harmless, simple folk, living chiefly by hunting; they lime birds, catch fish by poisoning the water, and are skilled in getting wild honey; they have bows with iron-pointed arrows and breed hunting dogs. They dwell in caves or bark huts. and their word for house is Sinhalese for a hollow tree, rukula. They count on their fingers, and make fire with the simplest form of fire-drill twirled by hand. They are monogamous, and their conjugal fidelity contrasts strongly with the vicious habits of the Sinhalese. Their religion has been described as a kind of demon worship, consisting of rude dances and shouts raised to scare away the evil spirits, whom they confound with their ancestors. The Veddahs are not to be confounded with the Rodiyas of the western uplands, who are a much finer race, tall, wellporportioned, with regular features, and speak a language said to be radically distinct from all the Aryan and Dravidian dialects current in Ceylon. There is, however, in Travancore, on the mainland, a low-caste “ Veda” tribe, nearly black, with wavy or frizzly hair, and now speaking a Malayalim (Dravidian) dialect (jagor), who probably approach nearer than the insular Veddahs to the aboriginal pre-Dravidian “ negrito ” element of southern India and Malaysia.

 VEDDER, ELIHU (1836–), American artist, was born in New York City on the 26th of February 1836. He studied under the genre and historical painter Tompkins H. Matteson (1813-1884), at Sherburne, N.Y., later under Picot, in Paris, and then, in 1857-61, in Italy. After 1867 he lived in Rome, making occasional visits to America. He was elected to full membership in the National Academy of Design, New York, in 1865. He devoted himself to the painting of genre pictures, which, however, attracted only modest attention until the publication, in 1884, of his illustrations to the Rubaiyat of Omar Khayyam; these immediately gave him a high place in the art world. Important decorative work came later, notably the painting symbolizing the art of the city of Rome, in the Walker Art Gallery of Bowdoin College, Maine, and the five lunettes (in the entrance hall) symbolical of government,