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k

While vitamin K supplementation is necessary for most people who suffer with the conditions listed above, there are some people who should avoid taking supplements in amounts above the RDA of 65 mcg unless directed by a doctor to do so.

22The best modern authorities are K.

10At Kiel, K.

10Potassium osmichloride, K 2 OsC1 6, is formed when a mixture of osmium and potassium chloride is heated in a current of chlorine, or on adding potassium chloride and alcohol to a solution of the tetroxide in hydrochloric acid.

11The atomic weight of the metal has been determined by K.

11Diamond color is also critical to Daniel K engagement rings: all stones must be of D or G color grades to be placed in Daniel K designs, a level of quality that less than 1 percent of the world's diamond supply can achieve.

11K.

00At Sodankyla, in 1882-1883, K.

00In Styria, according to K.

0016, 1899, p. 128; (84) K.

00Munchen, p. 187) and K.

00Wittmann, Monumenta Wittelsbacensia (Urkundenbuch, Munich, 1857-1861); K.

00k.

00Delbrtick and K.

00Waitz, Deutsche Verfassungsgeschichte, 3rd ed., revised by K.

00(Leipzig, 1889); K.

00Novodvorsky, The Struggle for Livonia, 1570-1582 (Rus.) (St Petersburg, 1904); K.

00Grossen (1892); K.

00Wallis Budge (1896, 2 vols., with English translation); the Syriac text of pseudo-Callisthenes by Budge (Cambridge, 1889); cp. K.

0021 3-33 2; K.

00Nowack (2nd ed., 1905), K.

00In 1904 K.

00Schonhals, Biographie des K.

00Roth, Ethnological Studies among the North-west-central Queensland Aborigines (London, 1897); Mrs K.

008vo (London, 1883); K.

00The cases of two, four and six squares had been given by K.

00Willer and K.

00Many lives of Tetzel have been published on the Protestant and on the Catholic side, the most recent being Korner's (1880), K.

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

00Tommasina, 8 K.

001° K.

00Tomaschek, "Die alten Thraker" in Sitzungsberichte der k.

00to 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.

00Stromberger, Bertold von Regensburg, der grässte Volksredner des deutschen Mittelalters (1877), K.

00g', k, k', 1, m, The Anthomedusa in form is generally deep, bell-shaped.

00k, Stomach.

0068, 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.

00k, Nectocalyces (swimming bells).

00Wilmans, Jahrbiicher des deutschen Reichs unter K.

00(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."

00(k) Excommunication was either greater or less.

00C. Warren, Buddhism in Translations (Cambridge, Mass., 1896); Mrs Rhys Davids, Buddhist Psychology (London, 1900); K.

00- K.)

001, K).

00Apical cell, p. Wall marking limit between the plerome k, initial segment of root-cap. P and the pleriblem Pb.

00k,o.

00E, epidermis; q, phellogen; 1, cells, and ~1, the pheliogen of the lenticel; k, cortical parenchyma, containing chlorophyll.

002, K).

00I, J, K).

00Sunden, "De tribunicia potestate a Lucio Sulla imminuta" in Skrifter utgifna of k.

00Carlsson, Beitrdge zur Kenntniss der Anatomie der Schwimmvogel; K.

00Museum, 1891; K.

00A 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.

00546 (1904); K.

00In 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.

00Solutions 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.

00Trithionic 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.

00The 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.

00Ten segments can be recognized - according to the studies of K.

00The 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.

00The morphology of the abdomen, ovipositor and genital armature is dealt with by K.

00France, lx., 1891); in the Bruchidae by Riley (Insect Life, iv., v., 1892-1893; and in the Strepsiptera (Stylopidae) by K.

00Townsend, The Great Schoolmen (1881); K.

00': 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?

00Soc. i.; K.

001896); C. K.

00As 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.

00A History of Russian Diplomacy under the Empress Elizabeth Petrovna, 1741-1762 (1899); The First Romanovs, 1613-1725 (1905); K.

00P. Thomsen, The Relation between Ancient Russia and Scandinavia and the Origin of the Russian State (London, 1877); the series of works by K.

00K.)

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

00For a non-partisan account of Xavier's work in the East, see K.

00(K.

00There is a Russian translation by Neviedomski (7 parts, Moscow, 1883-1886), and an Hungarian version of cc. 1-38 by K.

00Wiegand 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.

00The chief modern authorities are K.

00The best editions are those by K.

00Ettmiiller first applied Lachmann's ballad-theory to the poem (1841), and K.

00There are many translations of the epic into modern German, the best known being that of K.

002 So K.

00The most valuable branch is the oyster N,; : E, A i=De;I{1a iladelphia ', o K E .'

00And as in Hebrew, the six letters b g d k p t are aspirated when immediately preceded by any vowel sound.

00Among other distinguished Russian explorers in Asia, the names of Lessar, Annentkov (who bridged the Trans-Caspian deserts by a railway), P. K.

00Berghaus, Landbuch des Herzogtums Pommern (Berlin, 1865-1876); the Codex Pomeraniae diplomaticus, edited by K.

00Barthold, Geschichte von Rugen and Pommern (Hamburg, 1839-1845); K.

00Glycerin was discovered in 1779 by K.

00Ramsay, Angevin Empire (London, 1903); K.

00He 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.).

005, c. ecr k FIG.

00e, em, k, lx, n, int, ecr, to the edge of the free mantle-skirt, is the conical shell.

00Close 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.

00k, 1, p, J affords a means of test ing the conclusion that we have in Lankester's 4 capito-pedal bodies the rudimentary ctenidia.

00k antero-postero median section 6, 7, with following additions.

00Narrow process of the same running below the intestine and leading by k into the pericardium.

00K, Complete eversion of H.

00Beneath 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.

00k, 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.

00The latter opens e, Hermaphrodite duct into the common duct at the point k, (uterine portion).

0039, k).

00k, Retractor muscles.

00The Oncidiidae are, according to K.

00Kirchentag, and two years later founded and edited (1850-1861), with Neander and K.

00This assemblage is now generally regarded as a great division (phylum or sub-phylum) of the animal kingdom and known by K.

00The recent suggestion of K.

00k, Episternum.

0012, k), which have a renal function.

00k, Kidney tubes.

00These results were confirmed by the observations of K.

00lxxvi., 1904); K.

00- K.

00lxxiv., 1899); K.

00Wissens., Wien, lvi., 1889); K.

00Korschelt and K.

00lxix., 1897); K.

00et 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.

00That the different ranks or degrees of circular groups exhibited in the animal kingdom are Nine k in number, each being involved within the other."

00Among contem p orary writers in a more popular style are John Burroughs; Herbert K.

00Lastly, the square was extended southwards in the 16th century, when the new palace of the procurators, K, was built by Scamozzi.

00Bushnell..1896-1900George K.

00Graber, K.

00Again he heated fluorspar with oil of vitriol, as K.

00Schulze, August Neander (1890); and K.

00His collected works were translated into German and published by K.

00k n (S, -chunb a° Kwang`Chen ?,?z - `; r' 4; .

00.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.

00It 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.

004; K.

00k' b Ft.Myers un i!

00Its consonants are k, g, ng, ch, j, n, t, d, n, p, b, m, y, r, l, w, s, h.

00Final k and h are all but suppressed in the utterance.

00Stadt Basel (3 vols., 1906 sqq.); K.

00The name cerargyrite is a Greek form (from itpas, horn, and a pyvpos, silver) of the older name hornsilver, which was used by K.

00Ru2C16.4KC1; 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.

00In 1887 he was created C.M.G., and in 1888 K.

00He studied theology at Bonn (from 1822) under K.

00PANENTHEISM, the name given by K.

00Black, K.

00Priestley and K.

00Thus 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).

00The 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.

00Group I.: the alkali metals Li, Na, K, Rb, Cs, and also Ag, monovalent; Cu, monovalent and divalent; Au, monovalent and trivalent.

00These discoveries were followed by Daniel Rutherford's isolation of nitrogen in 1772, and by K.

00However, 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.

00The 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.

00To K.

00Progress 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.

00Eliminating 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.

00From the relation between the critical constants Pk Vk/Tk = 37 R or T k /P k = 3 .

007V k / R, and since Vk is proportional to the volume at absolute zero, the ratio T k /P k should exhibit additive relations.

00K= (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.

00According to the electromagnetic theory of light K = N2, where N is the refractive index for rays of infinite wave-length.

00The 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.

00Now 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.

00n 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°).

00Normal 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 .

00The ordinary structural formula of potassium sulphate is K - O - S - O - K.

00Such parallel overgrowths, termed episomorphs, are very common among the potassium and sodium felspars; and K.

00K, Na, Cs, Rb, Li; Tl, Ag.

00Jacobson, 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.

00K A1um=z00% K Alum= o/, Tl Alum= o% Tl Alum_ zoo FIG.

00K PO 4 = z00% NI-4112P04= x00% FIG.

00e, Retinaculum enclosing a k, Line of division between the nerve.

00Gundlach, Hesse and die Mainzer Stiftsfehde (Marburg, 1899); Walther, Literarisches Handbuch fir Geschichte and Landeskunde von Hesse (Darmstadt, 1841; Supplement, 1850-1869); K.

00In 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.

00Tiemann and K.

00Another Greek, Eumenes of Cardia, was chief secretary (apxtypa k uaTEUS).

00There seems to be even less chance for the combination of coloured strata and hachures proposed by K.

00The diphthong ai is 1 K.

00In 1 Facsimiles of it have been published by Desjardins(1869-1871), by K.

00minor K ? ?

00(y!K R.Fubsuy Man?

00Kunstmann's Entdeckung Amerikas (Munich, 1859), K.

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

00For the early stages of Kabbalistic theories, see K.

00by K.

00Gieseler, and Karl Immanuel Nitzsch for colleagues, he was called in 1827 to GÃ¶ttingen to succeed K.

00Knorre and K.

00Better results have attended the process of K.

00The next period was inaugurated in analysis by K.

00Mechanik in ihrer Entwickelung (Prague, 1883), English translation under the title, The Science of Mechanics (London, 1893) K.

00Ga Rakh Sofia Srule zi C.; Emine B ie C A lC K Sazopolis ?1.

00ry °Alessi Mata j `M Kadt K cui 0.

00I 90`ls K cr?wSao 3?t` rin l se vai?

00von Hahn (in " Denkschriften " of the K.

00Hercher (1873); see also K.

00Similar native iron has later been found by K.

00Nathorst, Otversigt of K.

00Storm, Studies on the " Vineland " Voyages (Copenhagen, 1889); Extraits des Memoires de la Societe Royale des Antiquaires du Nord (1888); K.

00Sillem (1903); Gallois, Geschichte der Stadt Hamburg (1853-1856); K.

00Grimm, Deutsche Rechtsalterthiimer (GÃ¶ttingen, 1828); K.

00xix.; K.

00Here 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.

00The 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.

00If 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.

00The 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.

00The 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.

00Thus 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).

00The 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.

00The 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.

00In 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.

00Like 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.

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

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

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

00j, k, 1, Lateral muscles (j, an 26, Interior of dorsal valve.

00teriors; k, middles; 1, g, Umbonal muscular impresoutsiders), enabling the sions (open valves).

00It 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.

00Let k such transpositions be necessary; then the expression X(kal aa2 N a 3.

00are called the elements of the determinant; the term (-) k alaa20a37...anv is called a member of the determinant, and there are evidently n!

00expression for the determinant becomes Z(-) k aitia2aa3y...anv, viz.

00From 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.

00a 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.

00Since 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 .>

00Similarly ali a21 a31 A =E a ik a2k a3k A li i > k > r, zï¿½k'r alr a2r air 23',!

00We 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.

00If 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.

00such 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.

00ï¿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 .

00Resultants.-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.

00R 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."

00CY 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.

00The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

00Discriminants.-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.

00It 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).

00Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

00Xic-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 = ...

00function 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 ..

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

00The 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.

00A 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.

00By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.

00+(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).

00an_1 i a n, and in general a n-k a 2 is the symbol for Q k.

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

00For 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?

00we may write (AB)i(AC)j(BC)k...

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

00Notice, therefore, that the symbolic product (ab)i(ac)j(bc)k...

00In order that (ab)i(ac)j(bc)k...

00possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

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

002 (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+...

00of a4) of a 4) k m!

00l 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.

00(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.

00n of k; if k = o we have the product of the two forms, and for all values of k>n the transvectants vanish.

00It is obvious that, when k is uneven, the kth transvectant of a form over itself does vanish.

00We 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.

00(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.

00The 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.

00k (DwAk) Ak 0; (D (ï¿½ A k) Ak =wJ.

00k k According to the well-known law for the changes of independent variables.

00Now D A xA k = (n - k) A k; Aï¿½ A k = k A?1; D ï¿½A A k = (n - k) A k+1;D mï¿½ A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = wJ; equations which are valid when X 1, X 2, ï¿½ 1, ï¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akï¿½ Hence °a-do +(n -1)71 (a2aa-+...

00% -k Y k = (af) k a n x.

00uo; 4, 1=0=f.; where u 0 =1, u1=o, assume that tfik = (af) k ay -k = f.

00u k =ï¿½y.

00ukx(n-2) ï¿½ Taking the first polar with regard to y (n - k) (a f) xa x -k-l ay+ k (af) k-l ay -k (ab) (n -1) b12by n kn-2k-1 n-1 k(n-2) =k(n- 2)a u x u5+nax ayux and, writing f 2 and -f l for y1 and 3,21 (n-k)(a f) k+ta i k-1 + k (n - 1)(ab)(a f) k-1 (b f)4 1 k by-2 = (uf)u xn-2k-1?

00Moreover the second term on the left contains (a f)' c -2b z 2 = 2 (a f) k-2b x 2 - (b) /0-2a 2 ï¿½ if k be uneven, and (af)?'bx (i f) of) '-la if k be even; in either case the factor (af) bx - (bf) ax = (ab) f, and therefore (n-k),bk+1 +Mï¿½f = k(n-2)f.(uf)uxn-2k-1; and 4 ' +1 is seen to be of the form f .14+1.

00take and Uk = (af) k ai k the linear factor which occurs to the second power in f.

00The discriminant, whose vanishing is the condition that f may possess two equal roots, has the expression j 2 - 6 i 3; it is nine times the discriminant of the cubic resolvent k 3 - 2 ik- 3j, and has also the expression 4(1, t') 6 .

00For, since -2t 2 =0 3 -21f 2, 6,-3j(-f) 3, he compares the right-hand side with cubic resolvent k 3 -21X 2 k - j 2.

00of f=0, :and notices that they become identical on substituting 0 for k, and -f for X; hence, if k1, k2, k 3 be the roots of the resolvent -21 2 = (o + k if) (A + k 2f)(o + k 3f); and now, if all the roots of f be different, so also are those of the resolvent, since the latter, and f, have practically the same discriminant; consequently each of the three factors, of -21 2, must be perfect squares and taking the square root 1 t = -' (1)ï¿½x4; and it can be shown that 0, x, 1P are the three conjugate quadratic factors of t above mentioned.

00We have A +k 1 f =0 2, O+k 2 f = x2, O+k3f =4) 2, and Cayley shows that a root of the quartic can be xpressed in the determinant form 1, k, 0.1y the remaining roots being obtained by varying 1, k, x the signs which occur in the radicals 2 u The transformation to the normal form reduces 1, k 3, ?

00If 4) = rx.sx, the Y2 =1 normal form of a:, can be shown to be given by (rs) 4 .a x 4 = (ar) 4s: 6 (ar) 2 (as) 2rxsy -I- (as) 4rx; 4) is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from J z+k 1 f =o, k 1 is any root of the resolvent.

00To determine them notice that R = (a6) and then (f, a 5) 5 = - R 5 (k1 +k2+k3) (f, a 4 5) 5 = - 5R5 (m 1 k 1+ m 2 k 2+ m 3 k 3), (f, a352) 5 = -10R5 (m21ke +m2k2+m3k3) three equations for determining k 1, k2, k3.

00l aa k -x 2 d d- = 0; Z(nk)ak+l adk - x ldd2=0; or in the form d d 52-x 2(7 =0, O - x1ax2 = 0; where 0 = ao d a l + 2a 1 -?...+na,,_id an, 0 = nal dao -?

00The general form of perpetuant is (4 K 3 A 2"`) and the generating function 1-z2.1-z3.1-z4 In general when 0 is even and =20, the condition is a l a 2 ...U 24 II(v 1 +a 2)II(a l +a 2 +cr 3)...II(Q 1 +a 2 -}-...

00(1-20) The actual form of a perpetuant of degree 0 has been shown by MacMahon to be +1 K0_1+1 K 3+20-4 K2, 01, 0-2, 0-3, ...3, 2), K 0, Ke -1, ...K 2 being given any zero or positive integer values.

00Or, lastly, we may leave the exponents h, k, j,1, untouched and consider the product i i i 2 .

00The alchemists designated it by the sign of Saturn k.

00A German translation of the last is contributed by K.

00A substance of which the real susceptibility is will, when surrounded by a medium having the susceptibility k', behave towards a magnet as if its susceptibility were - -}-4,rK').

00K is a commutator for reversing the direction of the magnetizing current, and G a galvanometer for measuring it.

009) from F to D, while at the same time the commutator K is rapidly worked, a series of alternating currents of gradually diminishing strength being thus caused to pass through the magnetizing coil.

00The current passes through the rocking key K, which, when thrown over to the right, places a in contact with c and b with d, and when thrown over to the left, places a in contact with e and b with f.

00When the switch S is closed, K acts simply as a commutator or current-reverser, but if K is thrown over from right to left while S is opened, not only is the current reversed, but its strength is at the same time diminished by the interposition of the adjustable resistance R2.

00The reversing key K having been put over to the left side, the short-circuit key S is suddenly opened; this inserts the resistance R, which has been suitably adjusted before hand, and thus reduces the current and therefore the magnetizing force to a known value.

00To continue the process, the key K is turned over to the right-hand side, and then, while S is open, is turned back, thereby not only reversing the direction of the current, but diminishing its strength by an amount depending upon the previous adjustment of R2.

00The experiments of K.

00Y Y' is a so.- iron yoke, which rocks upon knife-edges K and constitutes the beam of the balance.

00and K.

00Trans., 1888, 179A, 325), who found that the initial value of K was I.

00Nagaoka and K.

00Mag., 18 99, 1 7, 539; K.

00Verh., 1904, 6, 4, 21 I; K.

00Nagaoka, who also, in conjunction with K.

00The influence of temperature varying between wide limits has formed the subject of a research by K.

00Mag., 1898, 46, 261; 1902, 4, 45; K.

0013; K.

00Further contributions to the subject have been made by K.

00The tranverse electromotive force is equal to KCH/D, where C is the current, H the strength of the field, D the thickness of the metal, and K a constant which has been termed the rotatory power, or rotational coefficient.

00The following values of K for different metals are given by E.

00The subject, which is of importance in connexion with theories of magnetostriction, has been investigated by K.

00Owing to the difficulty of determining the magnetization I and the susceptibility K with accuracy, it has not yet been possible to submit this formula to a quantitative test, but it is said to afford an indication of the results given by actual experiment.

00K Xis'.

00The best modern determinations of the value of K for gaseous oxygen agree very fairly well with that given by Faraday in 18J3 (Exp. Res.

000, so that Io 6 K =33700/0.

00If V is the volume of a ball, H the strength of the field at its centre, and re its apparent susceptibility, the force in the direction x is f= K'VH X dH/dx; and if K',, and are the apparent susceptibilities of the same ball in air and in liquid oxygen, K' Q -K'o is equal to the difference between the susceptibilities of the two media.

00The magnetic properties of the metal at different temperatures and in fields up to 1350 units have been studied by P. Curie (loc. cit.), who found that its " specific susceptibility " (K) was independent of the strength of the field, but decreased with rise of temperature up to the melting-point, 273° C. His results appear to show the relation - K X10 6 = I'381 - O'o0155t°.

00Putting t°= - 182 in the equation given above for Curie's results, we get K X Io 6 = - 1.66, a value sufficiently near that obtained by Fleming and Dewar to suggest the probability that the diamagnetic susceptibility varies inversely as the temperature between-182° and the melting-point.

00de Phys., 18 95, 4, 204) of the specific susceptibility K of other diamagnetic substances at different temperatures.

00For all diamagnetic substances, except antimony and the value of K was found to be independent of the temperature.

00If W is the weight of iron present per c.c. at about io° C., then for ferric salts Io 6 K =266W-0'77 and for ferrous salts 10 6 K =206W - 077, the quantity - 0.77 arising from the diamagnetism of the water of solution.

00Annexed are values of Io 6 K for the different salts examined, w being the weight of the salt per c.c. of the solution.

00Sitz., 1897, 106, II.a, p. 623, and 1898, 107, II.a, p. 5) the atomic susceptibilities k of the metals nickel, chromium, iron, cobalt and manganese in solutions of their salts are as follows: - Fe(i) is iron contained in FeC1 2 and Fe(2) iron contained in Fe2(NOs)s.

00Curie has shown, for many paramagnetic bodies, that the specific susceptibility K is inversely proportional to the absolute temperature 0.

00k>00 6.

00Lindstrom, G., " Researches on the Visual Organs of the Trilobites," K.

00During the winter of 1916-7 these volunteers experienced heav y losses; after the Russian revolution in March 1917, Bolshevik sympathies spread among these troops and large sections of the people, while on the other hand national aspirations united the Farmers' Political League (40,000 members), headed by K.

0023, independence was declared, and K.

00Stern, on terrestrial magnetism by Goldschmidt, and on the method of least squares by K.

00k, Present corm.

00k', Younger corm produced from k.

00wh, Roots from k', which grows at expense of k.

00Young corm produced from k', in autumn, which in succeeding autumn will produce flowers.

00Kerp and K.

00over, following the indications already given by K.

00Potash alum, K 2 SO 4 ï¿½Al 2 (SO 4)aï¿½24H 2 O, crystallizes in regular octahedra and is very soluble in water.

00Chrome alum, K 2 SO 4 ï¿½Cr 2 (SO 4) 3.24H 2 O, appears chiefly as a by-product in the manufacture of alizarin, and as a product of the reaction in bichromate batteries.

00Pyevtsoff, K.

00Roborovsky and P. K.

00Scholl and K.

00Stier (1850), K.

00" J is aged 40, and K is aged 26; when will J be twice as old as K?

00Thus P = kQ+R, where k is an integer.

00Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t,.

00The symbol e 0 behaves exactly like i in ordinary algebra; Hamilton writes I, i, j, k instead of eo, el, e2, es, and in this notation all the special rules of operation may he summed up by the equalities = - I.

00xii.; K.

00If the primary wave at 0 be cos kat, the effect of the secondary wave proceeding from the element dS at Q is dS 1 dS - p cos k(at - p+ 4 A) = - -- sin k(at - p).

00If dS =27rxdx, we have for the whole effect 27r œ sin k(at - p)x dx, f P ' or, since xdx = pdp, k = 27r/A, - k fr' sin k(at - p)dp= [- cos k(at - p)]°° r.

00cos k(at-r), it is necessary to suppose that the integrated term vanishes at the upper limit.

00The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above, cos k(at-r 1)-cos k(at-r 2) =2 sin kat sin k(r1-r2), r2, r i being the distances of the outer and inner boundaries from the point in question.

002 2 c os k d x d y] .

00We readily find (with substitution for k of 27r/X) a2b S n J s in fl „2a2E2 „2b2n2 f2X2 f2X2 as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes.

00Writing for brevity k =p, k =q, (1), we have for the general expression (§ 11) of the intensity X2 f 212 = S 2 +C 2..

00f + 1 sin k a - f+ " dxdy = - 2 1h sin n k ?.

00k p h s i n h .

00s i n k Eh at - f - f ?

00We thus get for the disturbance at E, 7 t, due to this stream knl s in f 2f t f t .s i n k at - f - - { - 2 f.

00The original investigation of Stokes, here briefly sketched, extends also to the case where the streams are of unequal width h, k, and are separated by an interval 2g.

00Ascending series for C and S were given by K.

00(11), where r is the distance between the element dx dy dz and the point where a l is estimated, and k = n/b = 27r/X (12), X being the wave-length.

00See also for ethnological questions, Mongolian hypothesis: K.

00Iranian hypothesis: K.

00K i tchener an idea, and he resolved upon the scheme of fencing in areas by chains of blockhouses such as those already constructed for the protection of the railways.

00Hettema, jun., " Geschiedenis van het grondgebied der Zuid-Afrikaansche Republiek," Tijds k.

00Pernter (Neues fiber den Regenbogen, Vienna, 1888) and by K.

00K 2 SnO 3 into K2SnF6), which are closely analogous to, and isomorphous with, fluosilicates.

001 "dass einer der Sektierer von den andern als Christus verehrt werde," K.

00des k.

00Schirrmacher, Die letzten Hohenstaufen (GÃ¶ttingen, 1871); K.

00At present the best texts are those by K.

00Hozier, The Seven Weeks' War (1867; new edition, London, 1906); Antheil des k.

00(Leipzig, 1867); Bavarian General Staff, Antheil der k.

00Joachim, Die Politik des letzten Hochmeisters in Preussen, Albrecht von Brandenburg (Leipzig, 1892); K.

00Schultze, "Die altchristlichen Grabstatten Siziliens," Jahrbuch des k.

00Amongst special treatises may be mentioned K.

00ath ' K;idbrooke Hither Green: Wes i Bromley C Longitude West 0°4' of Greenwich Rai lways....-- underground Canals ' '; County Metropolitan Boroughs ...... :: ...

00a g s?o 3 oldeHr? ?k'?o r{d?El rpaoumc „14 ???(?

00Benzoflavin, an isomer of chrysaniline,1`is also a dye-stuff, and has been prepared by K.

00It is only soluble in a mixture of hydrofluoric and nitric acid, or in solutions of the caustic alkalis, in the latter case yielding hydrogen and a silicate: Si-}-2KHO+H 2 O = K 2 SiO 3 +2H 2.

00Potassium percarbonate, K 2 C 2 0 6, is obtained in the electrolysis of potassium carbonate at -10 to -15°.

00The five K's are (I) the kes or uncut hair of the whole body, (2) the kachh or short drawers ending above the knee, (3) the kara or iron bangle, (4) the khanda or small steel dagger,(5) the khanga or comb.

00For if the liquid of density a rises to the height h and of density p to the height k, and po denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be ah+Po and pk +po, and these being equal we have Uh = pk.

00(I) The principle is illustrated in the article Barometer, where a column of mercury of density a and height h, rising in the tube to the To:ricellian vacuum, is balanced by a column of air of density p, which may be supposed to rise as a homogeneous fluid to a height k, called the height of the homogeneous atmosphere.

00this makes k 27,200 ft., about 8300 metres.

00With uniform temperature, taking h constant in the gas-equation, dp / dz= =p / k, p=poet/ k, (9) so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p 1 and 1, 2 at heights z 1 and z2 (z1-z2)11?

00In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a, (1-1?-=(n+ I) Po --a 2 dr, (12) gravity varying inversely as the square of the distance r from the centre; so that, k = po/po, denoting the height of the homogeneous atmosphere at the surface, 0 is given by (n+I)k(I -9/6 0) =a(I -a/r), (13) or if c denotes the distance where 0=o, 0 _a (14) 0 r c -a' When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as p - po=k(P - Po), or P/po=I+(p-p0)/A, A=kpo, (15) so that A is the pressure due to a head k of the liquid at density under atmospheric pressure po; and it is the gauge pressure required on this law to double the density.

00Then dp/dz=kdp/dz = P, = Poe ik, p - po= kpo(ez Ik -1); (16) and if the liquid was incompressible, the depth at pressure p would be (p - po) 1po, so that the lowering of the surface due to compression is ke h I k -k -z= 1z 2 /k, when k is large.

00(17) For sea water, A is about 25,000 atmospheres, and k is then 25,000 times the height of the water barometer, about 250,000 metres, so that in an ocean 10 kilometres deep the level is lowered about 200 metres by the compressibility of the water; and the density at the bottom is increased 4%.

00(2) (3) Suppose the density p varies as some nth power of the depth below 0, then (7) and the lowering of the surface is 2 ° - z=klog po - z= - k log(1 - k) - zt12 k (20) Po as before in 17).

00Proceeding as in § 16 for the determination of the C.P. of an area, the same argument will show that an inclining couple due to K FIG.

00The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by P sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area.

00and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and satisfies the condition of (6) § 23; so that K is uniform throughout the fluid at any instant, and changes with the time only, and so may be replaced by F(t).

00d - K dK dK _ dK dK dK ?dx n dyd °, udx dz - ° and K=fdp/o+V+2q 2 =H (3) is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid.

00Thus if d,/ is the increase of 4, due to a displacement from P to P', and k is the component of velocity normal to PP', the flow across PP' is d4 = k.PP'; and taking PP' parallel to Ox, d,, = vdx; and similarly d/ ' = -udy with PP' parallel to Oy; and generally d4,/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.

00Over any ellipse n, moving with components U and V of velocity, =i+Uy-Vx=[msh(n-a) cos (3+Ucshn] sin k -[msh(n-a) sin (3+Vcchn] cos h; (7) so that ' =o, if U c sh n cos R, V = c ch n sin a, (8) m sh(n - a) m sh(n - a).

00The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

00Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.

00Thus, in (5), the cyclic constant k = 27rm.

00(to) Integrating over the base, to obtain one-third of the kinetic energy T, 3T = 2 pf '3 4R2(3x4-h4)dx/h 3 = pR2h4 / 1 35 V 3 (II) so that the effective k 2 of the liquid filling the trianglc is given by k 2 = T/Z p R 2 A = 2h2/45 = (radius of the inscribed circle) 2, (12) or two-fifths of the k 2 for the solid triangle.

00Trans., 1890); the solution is given by ch nS2=sn w, shnS2=i cn w (II) so that, round the boundary of the polygon, ik = K', sin n8 =o; and on the surface of the vortex 1P= o, q = Q, and cos n8=sn4p,nB= Zit -am sic, (12) the intrinsic equation of the curve.

00vertical, where Wh tan 0 = N = (c 2 - cl) c2 g2 tan 0, (6) (7) in which we have put k' 2 = ek 2, where E is a numerical factor depending on the shape.

00the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.