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k

The best modern authorities are K.

See K.

K.

At Sodankyla, in 1882-1883, K.

At Kiel, K.

In Styria, according to K.

16, 1899, p. 128; (84) K.

Munchen, p. 187) and K.

Wittmann, Monumenta Wittelsbacensia (Urkundenbuch, Munich, 1857-1861); K.

k.

(Vienna, 1854-1858); K.

Delbrtick and K.

Waitz, Deutsche Verfassungsgeschichte, 3rd ed., revised by K.

(Leipzig, 1889); K.

Novodvorsky, The Struggle for Livonia, 1570-1582 (Rus.) (St Petersburg, 1904); K.

Grossen (1892); K.

Wallis Budge (1896, 2 vols., with English translation); the Syriac text of pseudo-Callisthenes by Budge (Cambridge, 1889); cp. K.

21 3-33 2; K.

Nowack (2nd ed., 1905), K.

In 1904 K.

Schonhals, Biographie des K.

Roth, Ethnological Studies among the North-west-central Queensland Aborigines (London, 1897); Mrs K.

8vo (London, 1883); K.

The cases of two, four and six squares had been given by K.

Willer and K.

Many lives of Tetzel have been published on the Protestant and on the Catholic side, the most recent being Korner's (1880), K.

- Wireless Telegraphy The early attempts to achieve electric telegraphy involved the use of a complete metallic circuit, but K.

Tommasina, 8 K.

1° K.

Tomaschek, "Die alten Thraker" in Sitzungsberichte der k.

to allow him to dissolve parliament, entrusted Signor Giolitti, a Piedmontese deputy, sometime treasury minister in the Crispi cabinet, with the formation of a ministry of the Left, which contrived to obtain six months supply on account, and dissolved the Chamber, The ensuing general election (November 1892), marked by unprecedented violence and abuse of official pressure upon B k the electorate, fitly ushered in what proved to be scandals, the most unfortunate period of Italian history since the completion of national unity.

Stromberger, Bertold von Regensburg, der grässte Volksredner des deutschen Mittelalters (1877), K.

g', k, k', 1, m, The Anthomedusa in form is generally deep, bell-shaped.

k, Stomach.

68, k), absent in Chondrophorida and Cystophorida; they are contractile and resemble, both in appearance, structure and function, the umbrella of a medusa, with radial canals, ring-canal and velum; but they are without manubrium, tentacles or sense-organs, and are always bilaterally symmetrical, a peculiarity of form related with the fact that they are attached on one side to the stem.

k, Nectocalyces (swimming bells).

Wilmans, Jahrbiicher des deutschen Reichs unter K.

(k) In England the Constitutions of Clarendon added a provision for appeal to the king, " and if the archbishop shall have failed in doing justice recourse is to be had in the last resort (postremo) to our lord the king, that by his writ the controversy may be ended in the court of the archbishop; because there must be no further process without the assent of our lord the king."

(k) Excommunication was either greater or less.

C. Warren, Buddhism in Translations (Cambridge, Mass., 1896); Mrs Rhys Davids, Buddhist Psychology (London, 1900); K.

- K.)

1, K).

Apical cell, p. Wall marking limit between the plerome k, initial segment of root-cap. P and the pleriblem Pb.

k,o.

E, epidermis; q, phellogen; 1, cells, and ~1, the pheliogen of the lenticel; k, cortical parenchyma, containing chlorophyll.

2, K).

I, J, K).

Sunden, "De tribunicia potestate a Lucio Sulla imminuta" in Skrifter utgifna of k.

Carlsson, Beitrdge zur Kenntniss der Anatomie der Schwimmvogel; K.

Museum, 1891; K.

A favourite contrast for which there is more to be said is that drawn between the m k agico-religious spell-ritual, that says in effect, "My will be done," and the spirit of "Thy will be done" that breathes through the highest forms of worship. Such resignation in the face of the divine will and providence is, however, not altogether beyond the horizon of primitive faith, as witness the following prayer of the Khonds of Orissa: "We are ignorant of what it is good to ask for.

546 (1904); K.

In many cases it acts as a reducing agent (when used in the presence of acids); thus, permanganates are reduced to manganous salts, iodates are reduced with liberation of iodine, &c., 2KMnO 4 + 550 2 + 2H 2 0 = K 2 SO 4 + 2MnSO 4 + 2H 2 SO 4; 2K103+ 550 2 + 4H 2 O =1 3 + 2KHSO 4 + 3H2S04.

Solutions of persulphates in the cold give no precipitate with barium chloride, but when warmed barium sulphate is precipitated with simultaneous liberation of chlorine: K 2 S 2 0 8 + BaC1 2 = BaSO 4 + K 2 SO 4 + C1 2.

Trithionic acid, H2S306, is obtained in the form of its potassium salt by the action of sulphur dioxide on a solution of potassium thiosulphate: 2K 2 S 2 0 3 -f3S0 2 = 2K 2 S 3 0 6 -{- S; or by warming a solution of silver potassium thiosulphate KAgS 2 0 3 = Ag 2 S K 2 S 3 0 6; whilst the sodium salt may be prepared by adding iodine to a mixture of sodium thiosulphate and sulphite: Na 2 S0 3 -fNa 2 S 2 0 3 -f12 = Na 2 S 3062NaI.

The solution obtained may be evaporated in vacuo until it attains a density of 1.46 when, if partially saturated with potassium hydroxide and filtered, it yields crystals of potassium pentathionate, K 2 S 5 0 6.3H 2 0.

Ten segments can be recognized - according to the studies of K.

The chief points in the life-history of Stylops and Xenos, which are parasitic on certain bees (Andrena) and wasps (Polistes), have been investigated by K.

The morphology of the abdomen, ovipositor and genital armature is dealt with by K.

France, lx., 1891); in the Bruchidae by Riley (Insect Life, iv., v., 1892-1893; and in the Strepsiptera (Stylopidae) by K.

Townsend, The Great Schoolmen (1881); K.

': opal e ° °o T A R ple ' ag a ',ap iJ,wl Karkinit A C K r B L Scale, English Miles D S E A 32 Stavropol P O L A PI A N L A s E Derbent ° I?

Soc. i.; K.

1896); C. K.

As illustrating the general impoverishment of the Russian peasantry, it may be stated that the arrears of taxation owed by them have increased enormously since 1882, when they a, ounted to £2,854,000, until in 1900 the total amount was put k £15,222,000.

A History of Russian Diplomacy under the Empress Elizabeth Petrovna, 1741-1762 (1899); The First Romanovs, 1613-1725 (1905); K.

P. Thomsen, The Relation between Ancient Russia and Scandinavia and the Origin of the Russian State (London, 1877); the series of works by K.

K.)

8), and a rather larger distance in America k FIG.

For a non-partisan account of Xavier's work in the East, see K.

(K.

There is a Russian translation by Neviedomski (7 parts, Moscow, 1883-1886), and an Hungarian version of cc. 1-38 by K.

Wiegand and Schrader in 1895-1898 have laid bare the site of the Greek Priene, and the same has been done for the remains of Magnesia ad Maeandrum by French excavators in 1842-1843 and the German expedition under K.

The chief modern authorities are K.

The best editions are those by K.

Ettmiiller first applied Lachmann's ballad-theory to the poem (1841), and K.

There are many translations of the epic into modern German, the best known being that of K.

2 So K.

The most valuable branch is the oyster N,; : E, A i=De;I{1a iladelphia ', o K E .'

And as in Hebrew, the six letters b g d k p t are aspirated when immediately preceded by any vowel sound.

Among other distinguished Russian explorers in Asia, the names of Lessar, Annentkov (who bridged the Trans-Caspian deserts by a railway), P. K.

Berghaus, Landbuch des Herzogtums Pommern (Berlin, 1865-1876); the Codex Pomeraniae diplomaticus, edited by K.

Barthold, Geschichte von Rugen and Pommern (Hamburg, 1839-1845); K.

Glycerin was discovered in 1779 by K.

Ramsay, Angevin Empire (London, 1903); K.

He has only one symbol (written somewhat like a final sigma) for an unknown quantity, which he calls apc0µ6s (defined as "an undefined number of units"); the symbol may be a contraction of the initial letters ap, as A Y, K Y, D Y O, &c., are for the powers of the unknown (Suvaµcs, square; icu(30s, cube; Svva,uo& va i ccs, fourth power, &c.).

5, c. ecr k FIG.

e, em, k, lx, n, int, ecr, to the edge of the free mantle-skirt, is the conical shell.

Close to this the small renal organ (i, mediad) and the larger renal organ (k, to the right and posteriorly) are seen, also the pericardium (1) and a coil of the intestine (int) embedded in the compact liver.

k, 1, p, J affords a means of test ing the conclusion that we have in Lankester's 4 capito-pedal bodies the rudimentary ctenidia.

k antero-postero median section 6, 7, with following additions.

Narrow process of the same running below the intestine and leading by k into the pericardium.

K, Complete eversion of H.

Beneath the ciliated groove is placed an elongated ganglion (olfactory ganglion) connected by a nerve to the supraintestinal (therefore the primitively dextral) ganglion of the long h, k, m, Stomach.

k, Opening of the albuphrodite duct, which very soon becomes miniparous gland into P Y the hermaphrodite entwined in the spire of a gland - the duct.

The latter opens e, Hermaphrodite duct into the common duct at the point k, (uterine portion).

39, k).

k, Retractor muscles.

The Oncidiidae are, according to K.

Kirchentag, and two years later founded and edited (1850-1861), with Neander and K.

This assemblage is now generally regarded as a great division (phylum or sub-phylum) of the animal kingdom and known by K.

The recent suggestion of K.

k, Episternum.

12, k), which have a renal function.

k, Kidney tubes.

These results were confirmed by the observations of K.

lxxvi., 1904); K.

- K.

lxxiv., 1899); K.

Wissens., Wien, lvi., 1889); K.

Korschelt and K.

lxix., 1897); K.

et des todiers, which, though belonging to the same category as all the former, differs from them in its more scientific treatment of the subjects to which it refers; and, in 1808, K.

That the different ranks or degrees of circular groups exhibited in the animal kingdom are Nine k in number, each being involved within the other."

Among contem p orary writers in a more popular style are John Burroughs; Herbert K.

Lastly, the square was extended southwards in the 16th century, when the new palace of the procurators, K, was built by Scamozzi.

Bushnell..1896-1900George K.

Graber, K.

Again he heated fluorspar with oil of vitriol, as K.

Schulze, August Neander (1890); and K.

His collected works were translated into German and published by K.

k n (S, -chunb a° Kwang`Chen ?,?z - `; r' 4; .

.o K il -ju tong-chin A B Longitude East 14 of Greenwich C 8° 3: E 44 legend, were bathing one day in a lake under the Chang-pai-Shan mountains when a passing magpie dropped a ripe red fruit into the lap of one of them.

It is found most convenient to make use of the sag of the wire produced when it is stretched between two fixed points (K 1 K 2, fig.

4; K.

k' b Ft.Myers un i!

Its consonants are k, g, ng, ch, j, n, t, d, n, p, b, m, y, r, l, w, s, h.

Final k and h are all but suppressed in the utterance.

Stadt Basel (3 vols., 1906 sqq.); K.

The name cerargyrite is a Greek form (from itpas, horn, and a pyvpos, silver) of the older name hornsilver, which was used by K.

Ru2C16.4KC1; Ru 2 C1 6.4NH 4 C1, &c. The pure tetrachloride, RuC1 4, has not been isolated, but is chiefly known in the form of its double salts, such as potassium ruthenium chloride, K 2 RuC1 6, which is obtained when finely divided ruthenium is fused with caustic potash and potassium chloride is gradually added to the fused mass (U.

In 1887 he was created C.M.G., and in 1888 K.

He studied theology at Bonn (from 1822) under K.

PANENTHEISM, the name given by K.

Black, K.

Priestley and K.

Thus copper sulphate was CuO+S0 3, potassium sulphate 2S0 3 +P00 2 (the symbol Po for potassium was subsequently discarded in favour of K from kalium).

The metals may be arranged in a series according to their power of displacing one another in salt solutions, thus Cs, Rb, K, Na, Mg, Al, Mn, Zn, Cd, Tl, Fe, Co, Ni, Sn, Pb, (H), Sb, Bi, As, Cu, Hg, Ag, Pd, Pt, Au.

Group I.: the alkali metals Li, Na, K, Rb, Cs, and also Ag, monovalent; Cu, monovalent and divalent; Au, monovalent and trivalent.

These discoveries were followed by Daniel Rutherford's isolation of nitrogen in 1772, and by K.

However, in 1833, Berzelius reverted to his earlier opinion that oxygenated radicals were incompatible with his electrochemical theory; he regarded benzoyl as an oxide of the radical C 14 H 1Q, which he named " picramyl " (from 7rucp6s, bitter, and &uvyalk, almond), the peroxide being anhydrous benzoic acid; and he dismissed the views of Gay Lussac and Dumas that ethylene was the radical of ether, alcohol and ethyl chloride, setting up in their place the idea that ether was a suboxide of ethyl, (C2H5)20, which was analogous to K 2 0, while alcohol was an oxide of a radical C 2 H 6; thus annihilating any relation between these two compounds.

The rapidity of the method, and the accurate results which it gave in the hands of a practised experimenter, led to its systematization by Jens Jakob Berzelius and Johann Friedrich Ludwig Hausmann, and in more recent times by K.

To K.

Progress in forensic chemistry was only possible after the reactions of poisons had been systematized; a subject which has been worked out by many investigators, of whom we notice K.

Eliminating a and b between these relations, we derive P k V k /Tk= 8R, a relation which should hold between the critical constants of any substance.

From the relation between the critical constants Pk Vk/Tk = 37 R or T k /P k = 3 .

7V k / R, and since Vk is proportional to the volume at absolute zero, the ratio T k /P k should exhibit additive relations.

K= (I +2a)/(I -a), or a=(K-I)/(K+2), where K is the dielectric constant and a the fraction of the total volume actually occupied by matter.

According to the electromagnetic theory of light K = N2, where N is the refractive index for rays of infinite wave-length.

The relation they suspected to be of the form -yS = KT, where K is a constant analogous to R, and S the surface containing one gramme-molecule, y and T being the surface tension and temperature respectively.

Now the value of K, -y being measured in dynes and M being the molecular weight of the substance as a gas, is in general 2.121; this value is never exceeded, but in many cases it is less.

n is the mean number of molecules which associate to form one molecule, then by the normal equation we have y (Mnv) 3 =2.121(r -6°); if the calculated constant be K 1, then we have also y(Mv)3=K,(r-6°).

Normal values of K were given by nitrogen peroxide, N204, sulphur chloride, S 2 C1 21 silicon tetrachloride, SiC1 4, phosphorus chloride, PC1 3, phosphoryl chloride, POC1 31 nickel carbonyl, Ni(CO) 4, carbon disulphide, benzene, pyridine, ether, methyl propyl ketone; association characterized many hydroxylic compounds: for ethyl alcohol the factor of association was 2.74-2.43, for n-propyl alcohol 2.86-2.72, acetic acid 3.62 -2.77, acetone 1 .

The ordinary structural formula of potassium sulphate is K - O - S - O - K.

Such parallel overgrowths, termed episomorphs, are very common among the potassium and sodium felspars; and K.

K, Na, Cs, Rb, Li; Tl, Ag.

Jacobson, Lehrbuch der organischen Chemie; Richter-Anschutz, Organische Chemie (I I th ed., K 2 S04 K2 S04 (NH 4) 2 SO 4 =o% SO 4 = coo FIG.

K A1um=z00% K Alum= o/, Tl Alum= o% Tl Alum_ zoo FIG.

K PO 4 = z00% NI-4112P04= x00% FIG.

e, Retinaculum enclosing a k, Line of division between the nerve.

Gundlach, Hesse and die Mainzer Stiftsfehde (Marburg, 1899); Walther, Literarisches Handbuch fir Geschichte and Landeskunde von Hesse (Darmstadt, 1841; Supplement, 1850-1869); K.

In Phoenician itself and in the other Semitic alphabets the position of the middle legs of the W is altered so that the symbol takes such forms as or V or w, ultimately ending sometimes in a form like K laid sideways, he.

Tiemann and K.

Another Greek, Eumenes of Cardia, was chief secretary (apxtypa k uaTEUS).

There seems to be even less chance for the combination of coloured strata and hachures proposed by K.

The diphthong ai is 1 K.

In 1 Facsimiles of it have been published by Desjardins(1869-1871), by K.

minor K ? ?

(y!K R.Fubsuy Man?

Kunstmann's Entdeckung Amerikas (Munich, 1859), K.

'Eav aµap T'n0" [f 6 aSEXIpOS UOv K aT v, TiUE6 ELp'nv7J iS7rayE g AE'Y0v a6TOv vOv..

For the early stages of Kabbalistic theories, see K.

by K.

Gieseler, and Karl Immanuel Nitzsch for colleagues, he was called in 1827 to GÃ¶ttingen to succeed K.

Knorre and K.

Better results have attended the process of K.

The next period was inaugurated in analysis by K.

Mechanik in ihrer Entwickelung (Prague, 1883), English translation under the title, The Science of Mechanics (London, 1893) K.

Ga Rakh Sofia Srule zi C.; Emine B ie C A lC K Sazopolis ?1.

ry °Alessi Mata j `M Kadt K cui 0.

I 90`ls K cr?wSao 3?t` rin l se vai?

von Hahn (in " Denkschriften " of the K.

Hercher (1873); see also K.

Similar native iron has later been found by K.

Nathorst, Otversigt of K.

Storm, Studies on the " Vineland " Voyages (Copenhagen, 1889); Extraits des Memoires de la Societe Royale des Antiquaires du Nord (1888); K.

Sillem (1903); Gallois, Geschichte der Stadt Hamburg (1853-1856); K.

Grimm, Deutsche Rechtsalterthiimer (GÃ¶ttingen, 1828); K.

xix.; K.

Here the ions are potassium and the group Ag(CN)2.1 Each potassium ion as it reaches the cathode precipitates silver by reacting with the solution in accordance with the chemical equation K--+KAg(CN) 2 =2KCN+Ag, while the anion Ag(CN) 2 dissolves an atom of silver from the anode, and re-forms the complex cyanide KAg(CN) 2 by combining with the 2KCN produced in the reaction described in the equation.

The tests for a salt, potassium nitrate, for example, are the tests not for KNO 3, but for its ions K and NO 3, and in cases of double decomposition it is always these ions that are exchanged for those of other substances.

If µ be the molecular conductivity, and its value at infinite dilution, the fractional number of molecules dissociated is k /µop, which we may write as a.

The number of undissociated molecules is then I - a, so that if V be the volume of the solution containing I gramme-molecule of the dissolved substance, we get q= and p= (I - a)/V, hence x(I - a) V =yd/V2, and constant = k.

a V(I - a) This constant k gives a numerical value for the chemical affinity, and the equation should represent the effect of dilution on the molecular conductivity of binary electrolytes.

The equation then becomes a 2 /V = k, or a = A / Vk, so that the molecular conductivity is proportional to the square root of the dilution.

Thus in the case of cyanacetic acid, while the volume V changed by doubling from 16 to 1024 litres, the values of k were 0.00 (37 6, 373, 374, 361, 362, 361, 368).

The mean values of k for other common acids were - formic, 0.0000214; acetic, o 0000180; monochloracetic, 0.0.0155; dichloracetic, 0.051; trichloracetic, 1.21; propionic, 0.0000134.

The value of k, however, does not keep constant so satisfactorily in the case of highly dissociated substances, and empirical formulae have been constructed to represent the effect of dilution on them.

In dilute solution such substances as hydrochloric acid and potash are almost completely dissociated, so that, instead of representing the reaction as HC1+KOH = KC1 d-H20, we must write The ions K and Cl suffer no change, but the hydrogen of the acid and the hydroxyl (OH) of the potash unite to form water, which is only very slightly dissociated.

Like other plateaus, the great plateau of the centre of Asia, stretching from the Himalayas to Bering Strait, 2 has on its surface a number of gentle eminences (angehaufte Gebirge of K.

(K,Li)3[Al(OH,F)2]FeAl2S15016 Biotite.

(H,K)2(Mg,Fe)2(A1,Fe)2(S104) 3 Phlogopite..

[H,K, (MgF)13Mg3A1(S104)3 The water which is present in muscovite.

j, k, 1, Lateral muscles (j, an 26, Interior of dorsal valve.

teriors; k, middles; 1, g, Umbonal muscular impresoutsiders), enabling the sions (open valves).

It was long a matter in discussion whether the animal could displace its valves sideways when about to open its shell, but this has been actually observed by Professors K.

Let k such transpositions be necessary; then the expression X(kal aa2 N a 3.

are called the elements of the determinant; the term (-) k alaa20a37...anv is called a member of the determinant, and there are evidently n!

expression for the determinant becomes Z(-) k aitia2aa3y...anv, viz.

From the value of A we may separate those members which contain a particular element a ik as a factor, and write the portion aik A ik; A k, the cofactor of ar k, is called a minor of order n - i of the determinant.

a ll a33 ï¿½ï¿½ï¿½ a32 a33 ï¿½ï¿½ï¿½ a3n an2 an3 ï¿½ï¿½ï¿½ ann Similarly A ik, the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column.

Since the determinant having two identical rows, and an3 an3 ï¿½ï¿½ï¿½ ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+ï¿½ï¿½ï¿½ +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+ï¿½ï¿½ ï¿½ +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+ï¿½ï¿½ï¿½ +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+ï¿½ï¿½ï¿½+aniAnk = 0 (k .>

Similarly ali a21 a31 A =E a ik a2k a3k A li i > k > r, zï¿½k'r alr a2r air 23',!

We may say that, in the resulting determinant, the element in the ith row and k th column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklb11+ak2b12+ak3b13ï¿½ï¿½ï¿½ +aknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.

If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +ï¿½ï¿½ï¿½ +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.

such that Aik=Aki, for the determinant got by suppressing the ith row and k th column differs only by an interchange of rows and columns from that got by suppressing the k th row and i th column.

ï¿½ Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,ï¿½ï¿½ï¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .

Resultants.-When we are given k homogeneous equations in k variables or k non-homogeneous equations in k - i variables, the equations being independent, it is always possible to derive from them a single equation R = o, where in R the variables do not appear.

R is a function of the coefficients which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination."

CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.

The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).

Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

Xic-1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1= X 2 = ...

function of separations of (li'12 2 13 3 ...) of specification (si 1 s 22 s 33) Suppose the separations of (11 1 13 2 1 3 3 ...) to involve k different specifications and form the k identities ï¿½1s ï¿½ s Al A 2 A3 ..

(S - 1, 2, ...k), where (mï¿½lsm"`2sm"`38...) is one of the k specifications.

The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that () St =ti ts.

A table may be formed expressing the k expressions Pa l), P(2),...P(1) as linear functions of the k expressions (m"`'smï¿½2smï¿½3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry.

By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.

+(m -3) D 5(213) (214) (15) - (13) (14) (14), as= and and we see further that (alai +a2a2+...+amam) k vanishes identically unless (mod m).

an_1 i a n, and in general a n-k a 2 is the symbol for Q k.

= a k; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.

For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1ï¿½1 +a2m2) = aa a ï¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alï¿½1+a2ï¿½2) = a A ï¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a ï¿½ respectively.

we may write (AB)i(AC)j(BC)k...

(ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the invariant property.

Notice, therefore, that the symbolic product (ab)i(ac)j(bc)k...

In order that (ab)i(ac)j(bc)k...

possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

= t) 1 v ...axbxcx..., and assert that the symbolic product (ab)i(ac)'(bc)k...aibxc2...

2 (ac)(bc)anx xibn-i -1 x = (bc)2anbn-2Cn-2 + (ac)2an x x x The weight of a term ao°a l l ...an n is defined as being k,+2k2+...

of a4) of a 4) k m!

l ax 2 2 ax i l This is called the kth transvectant of f over 4); it may be conveniently denoted by (f, (15)k.

(a m b n) k (ab) kamkbn-k x, x - x it is clear that the k th transvectant is a simultaneous covariant of the two forms.

n of k; if k = o we have the product of the two forms, and for all values of k>n the transvectants vanish.

It is obvious that, when k is uneven, the kth transvectant of a form over itself does vanish.

We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the process as merely substituting (ab) k, (OW for azbx, 4x t ' respectively in the symbolic product subjected to transvection.

(f, 4)) k +(f, 4)) k +ï¿½(f, 4/) k +aï¿½(1, 4)')k; and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the k th transvectant of the separate products.

The process of transvection is connected with the operations 12; for?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of ax, viz.

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