# X sentence example

x

- Press Square, Square, Square, X, Square, Circle, Circle, R1 + X, Circle, R1 + Square, R1 + Square, R1 + Square, R1 + X, Circle, R1 + X, R1 + Square, R1 + X, Circle, R1 + X, R1 + Square, Circle, Square, X, Circle, Square, X, then Square.
- When symptoms continue even after treatment or to rule out the presence of other illnesses with similar symptoms, the diagnostic evaluation may include blood tests, a hydrogen breath test, or an x ray of the bowel, called a barium enema.
- She's no longer Madam X.
- Choices for your backsplash may include 12-inch marble or onyx tiles lit from behind, 6 x 12-inch glass tiles set in a running bond, or 18-inch metallic finish tiles that fill the space with only one vertical grout line every 18-inches.
- According to the Child Development Institute, it has been shown that children with Fragile X who are integrated into "normalized" educational and social environments "achieve above the level that would have been predicted from measured IQ".Advertisement
- First Twilight, then that TV guy, X.
- Leo X.
- But you were right—'X' equals a period!
- Don't drink the wine and don't let the cat in X's room, Jessi read aloud.
- Rule two, escort X's girls out every morning.Advertisement
- X's woman just pulled the wool over his eyes.
- x.
- He lectured at Padua, Naples, Rome and Pisa, and won so high a reputation that he was deputed by Leo X.
- He returned to his lawful allegiance in the following year and assisted Czarniecki in his difficult task of expelling Charles X.
- It was under Leo X.Advertisement
- As patron of learning Leo X.
- More recent studies have served to produce a fairer and more honest opinion of Leo X.
- - The life of Leo X.
- Roscoe, Life and Pontificate of Leo X.
- X.Advertisement
- 9, but only its mode of attachment (X, fig.
- Pausanias (x.
- A third form was employed in the case of the concordat of 1516 between Leo X.
- The situation was modified by the concordat of Bologna, which was personally negotiated by Leo X.
- The central feature of the estate is a château (375 X 150 ft.) of French Renaissance design, after the famous chateau at Blois, France.Advertisement
- 45.4 x.
- to x.
- They're calling you Girl X.
- had unanimously condemned, and the confirmation of the concordat between Leo X.
- stayed the Lutheran movement, and Luther himself, safe in the solitude of the Wartburg, survived Leo X.Advertisement
- Nitti, Leone X.
- Consider the particles which occupy a thin stratum dx perpendicular to the primary ray x.
- Kurz and X.
- Table X.
- (1897) 5 seq., x.Advertisement
- In some impor X XVII.
- x.) is the most important; for the first years of Frederick's reign see M.
- Kyaukpyu is a port under the Indian Ports Act (X.
- He has alluded to a childish fancy for a young girl with a slight obliquity of vision; but he only mentions it 1 Ouvres, x.
- In January 1274 he was summoned by Pope Gregory X.Advertisement
- Tidander, Kriget mellan Sverige och Ryssland dren 1555-1557 (Vesteras, 1888) P. Pierling, Un Arbitrage pontifical au X VI e siecle entre la Pologne et la Russie (Bruxelles, 1890); V.
- Tours, X.
- When cyanogen is prepared by heating mercuric cyanide, a residue known as para-cyanogen, (CN)x, is left; this is to be regarded as a polymer of cyanogen.
- 272) estimate of 470,000 as the 1 Pericles called Aegina the "eye-sore" (X) of the Peiraeus.
- the barbarians," says Strabo (x.Advertisement
- X XIII.
- In the Phaedrus (245 c) the argument is, that the soul is self-moving, and, therefore, immortal; and this argument is repeated in the Laws (x.
- 29; Mark x.
- 1) and with the good Samaritan (Luke x.
- is far worthier of admiration than either Charles X.Advertisement
- Thus, if x= horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraical symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers.
- Thus, 1 - x would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 - x) (1 - y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.
- 7ruppos, flamecoloured, and o-i&17Aos, iron): a scaly-fibrous variety from the same locality is called lepidocrocite (from X€iris, scale, and KpoKcis, fibre).
- He published Lives of Foreign Statesmen (1830), The Greek and the Turk (1853), and Reigns of Louis X VIII.
- and Charles X.
- In 1520 he went to Rome, where he entered the brilliant literary circle of Leo X.
- The mean elevation of the X XVII.
- It has a fine Renaissance facade, constructed about 1500 by Cardinal Giovanni de' Medici (afterwards Pope Leo X.), and some good terra cottas by the Della Robbia.
- Firth (1902); The Diplomatic Relations between Cromwell and Charles X.
- Dante, in the Paradiso- (x.
- He had agreed with Pope Leo X.
- Half the proceeds in his province were to go to him, half to Leo X.
- 1853) Le Reveil du sentiment religieux en France au X Vil e siecle, by Strowski (Paris, 1898); Four Essays on S.
- x. ?).
- Sclerostomum armatum, y, X about 31, opened to show the phagocytic organs.
- Owing to the high price of gutta-percha the tendency, of recent years, has been to approximate more closely to the theoretical dimensions, x xvl.
- In all cases of wave motion the wave-length is connected with the velocity of propagation of the radiation by the relation v=nX, where n is the frequency of the oscillations and X is the wave-length.
- 1 p, G r, 2 s, X t and d, V u and o, 8 f, d s (i.e.
- 1; Pausanias x.
- This page gives an overview of all articles in the 1911 Brittanica which are alphabetized under X to Yve.
- In Scotland Pope Gregory X.
- Rome, X.
- 1 These ceased to have legal currency at the end of igoi; they were notes of x and 2 lire.
- When Leo X.
- Meanwhile Leo X.
- died, and that same Cardinal Sarto became pope under the style of Pius X.
- Chapter X.
- Aulard, Les Portraits litteraires a la fin du X VIII" siecle, pendant la Revolution (Paris, 1883).
- opening closed by a plug of protoplasm (x, fig.
- cn, N, o.c, x, b, the 1.0¦6 From Gegenbaur's Elements of Comparative Anatomy.
- Morphology, x.
- Soc. x.
- Imp. Sci.) St Petersbourg (8) x.
- cap. x.
- 2 Considerations sur les corps organises, chap. x.
- In 1516 a Concordat between Leo X.
- In the 12th century the Church's rule, that subsequent marriage did legitimize previous issue, was settled (c. 6, x.
- On the re-establishing of the Catholic religion on the basis of the new Concordat, promulgated 18 Germinal, year X.
- exactly 8 X 5 days), now universal in the Eastern Church, originated in the 7th century.
- F.; X.)
- c/i, epidermis; st stoma; me,, mesophyil; pal, palisade; spa, spongy tissue; Isp, inteicellular space; wi., water tissue; x, xylem; p/i, phioem; Phil, phloeoterma; sri, scierenchyma.
- mesocyde; x.
- ,,,f.e \\~0.\~.~J \\ y~ \ \j/f / ~/ X 1W!
- The swellings have been found to be due to a curious hypertrophy of the tissue of the part, the cells being filled with an immense number of minute bacterium-like organisms of V, X or Y shape.
- (1896), x.
- 18; Rosenberg, Cytologische und morphologische Studien an Drosera ion gifolia X.
- Leo travelled extensively in the north and west of Africa, and was eventually taken by pirates and sold to a master who presented him to Pope Leo X.
- 76, x.
- Koirpos, dung, and X LOo, stone), the fossilized excrements of extinct animals.
- - End view of skull of a Chicken fo three weeks old, X 8 diameters.
- hawk (Accipiternisus), palatal view, X 2 diameters.
- - Os hyoides of adult Fowl, X II diameters.
- - Sternum of a Chick (Gallus domesticus) three days old, lower view, X three diameters.
- In many birds the insertion is shifted from the femur to the neck of the tibia, in which case the " accessory head " is said to be absent, a condition expressed by Garrod by the symbol X.
- X, Caud-ilio-flexorius.
- The most primitive combination, ambiens and A B X Y, is the most common; next follows that of A X Y, meaning the reduction of B, i.e.
- the iliac portion of the caud-ilio-femoralis; A B X and B X Y are less common; A X and X Y are rare and occur only in smaller groups, as in subfamilies or genera; B X occurs only in Podiceps.
- Further, the combinations B X Y and A X Y cannot be derived from each other, but both directly from A B X Y in two different directions.
- Keeping this in mind, we may fairly conclude that the flamingo with B X Y points to an ancestral condition A B X Y, which is still represented by Platalea and Ibis, whilst the other storks proper have taken a different line, leading to A X Y.
- Auditory ment in the crocodile, and with the ", chain " of Chicken, X 6 processus folii of the mammalian diameters; lateral and basal malleus, it follows that the whole views.
- Io, x.
- x.).
- ix., x.); where the champions of David fought those of Ish-bosheth (2 Sam.
- It was destroyed by Joshua for joining the league against the Gibeonites (Joshua x 3 1 -33) and assigned to the tribe of Judah (xv.
- in that time, or is moving at the rate of 5 X 6080 ft.
- Group of spore-cases (sorus) on back of leaf (X 4).
- 20-27; x.
- GREGORY X.
- The registers of Gregory X.
- Gregorio X, papa (Piacenza, 1876); F.
- (Innsbruck, 1895); A Zisterer, Gregor X.
- Walter, Die Politik der Kurie enter Gregor X.
- Loserth, "Akten tither die Wahl Gregors X."
- von Hirsch-Gereuth, "Die Kreuzzugspolitik Gregors X."
- Catherine, (VII.) Anne Cesarevich Ale x ius Anna, (IX.) Elizabeth duchess of (1730-40).
- 1,128 28,485 1,211 25,975 The casualties enumerated in items I, 4 and 7 of Table X.
- Items 2 and 5 in Table X.
- analyses the classes of accident comprised in items, 3 and 6 of Table X.
- per lb; then the mechanical energy available in footpounds per hour is approximately 0-06 X 778 X Ec, and this expressed in horse-power units gives I.H.P. - o 06X778XEc _648.1,980,000.
- per hour up a gradient of I in 300, the extra horse-power required will be H.P. _280X2240X58.6 =22 300 X 550 3.
- Therefore the horse-power which must be developed in the cylinders to effect this change of speed is from (21) H.P.280X2240X0 113X59 = _237 55 0 X 32 The rate of working is negative when the train is retarded; for instance, if the train had changed its speed from 41 to 40 m.
- Then the tractive force is, from (25), (149 X 19 2 X2.166)/6.25 =18,600lb =8.3 tons.
- ft., that the rate of combustion is 150 lb of coal per square foot of grate per hour, that the calorific value is 14000, and finally that n =0.06, the maximum indicated horse-power which the engine might be expected to develop would be o 06 X 150 x14000 X24 X 778/1980000 = I 190, corresponding to a mean effective pressure in the cylinders of 59.5 lb per square inch.
- 18, 19), or the " rod of His wrath," for the chastisement of Israel (x.
- But the instrument unduly exalts itself, and Assyria itself shall suffer humiliation at the hands of the world's divine sovereign (x.
- Job x.
- Innocent X.
- 53, 2 99, and x.
- 6 According to the most trustworthy accounts, but see Letters and Papers, x.
- 53, 2 99, x.
- 45 62 -4573; British School Annual, x.
- 1177, note 1) considers it " certain " that 'Axaia ='A X e wia, although he is unable to explain the form.
- It is mentioned by Isaiah (x.
- (1268) and the election of Gregory X.
- zero for carbon and oxygen, and x for carbon dioxide, we obtain the equation o+o=x+94300 cal.
- or x = - 94300 cal.
- Thus if we wish to ascertain the thermal effect of the action Mg+CaO =MgO+Ca, we may write, knowing the heats of formation of CaO and Mg0 to be 131000 and 146000 respectively, 0-131000 = 0-146000+x x =15000 cal.
- The above equation may consequently be written, if x is the heat of formation of methane, -x+0 = -94300-(2 X 68300) +213800 x =17000 cal.
- It seems to have been suggested in 1516, and although certain charters have been appealed to in proof of an earlier use of the title, it was first conferred by Pope Leo X.
- for the number i Kings x.
- Moreover, of the various accounts of the massacre of the princes of Judah, the Judaean ascribes it not to Jehu and the reforming party (2 Kings x.
- They entered into an agreement to obey its teaching, undertaking in particular to avoid marriages with foreigners (x.
- expelled the Jews from France, nine years later Louis X.
- 'Invous X ptar6s, Oeou `Tuffs, 16 y TIJp, Jesus Christ, Son of God, Saviour, which together spell the Greek word for "fish," ix9vs.
- His grandfather, Antoine Louis Marie, duc de Gramont (1755-1836), had emigrated during the Revolution, and his father, Antoine Heraclius Genevieve Agenor (1789-1855), duc de Gramont and de Guiche, fought under the British flag in the Peninsular War, became a lieutenantgeneral in the French army in 1823, and in 1830 accompanied Charles X.
- PTOLEMY X.
- and x.
- 5 with x.
- 27 makes Nahash himself David's ally, and accounts for David's eagerness to repay to Hanun, the son of Nahash, the kindness which he had received from the father (x.
- P.-P.; X.)
- 59, x.
- - X I I I, XIV, thirteenth the intestine.
- P. W.; X.)
- in 1514, and reaped his reward in the bishoprics of Lincoln and Tournai, the archbishopric of York, which was conferred on him by papal bull in September, and the cardinalate which he had sent Polydore Vergil to beg from Leo X.
- ELEANOR OF AQUITAINE (c. 1122-1204), wife of the English king Henry II., was the daughter and heiress of Duke William X.
- But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x.
- With one symbol for an unknown, it will easily be understood what scope there is foradroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined.
- Often assumptions are made which lead to equations in x which cannot be solved "rationally," i.e.
- Sometimes his x has to do duty twice, for different unknowns, in one problem.
- In general his object is to reduce the final equation to a simple one by making such an assumption for the side of the square or cube to which the expression in x is to be equal as will make the necessary number of coefficients vanish.
- 3, Cups enlarged X 5.
- And when he hath mowen his medowe, then he hath his medowe grounde, soo that if he hath any weyke catell that wold be amended, or dyvers maner of catell, he may put them in any close he wyll, the which is a great advantage; and if all shulde lye commen, than wolde the edyche of the come feldes and the aftermath of all the medowes be eaten in X.
- Although enormous single crops of mangels [[Table X]].--Decennial Average Yields in Great Britain of Wheat, Barley and Oats-Bushels per acre.
- Adopting this course in the case of the cereal crops of Great Britain the decennial averages recorded in Table X.
- Cyzicenus (reigned 116-95), the son of Cyzicenus, Antiochus X.
- P.; X.)
- lvg, Primarily left (subsequently x, x', Pins fastening the elastic the sub-intestinal) visceral cord (representing the vis ganglion.
- (Lankester.) x, y, The median antero-posterior axis.
- c x c ' 'Il" a r-v n c ?
- x, Glandular lamellae of the inner face of the mantle-skirt.
- The surface x of the mantle between the rectum and the gill-plume is thrown into folds which in many sea-snails (whelks or Buccinidae, &c.) are very strongly developed.
- x, Auricle of the heart.
- Bathanalia, from x, Filiform appendage (?
- x, Visceral ganglion.
- (3), x.
- Their chief man of action was a sturdy Breton peasant, Georges Cadoudal, whose zeal and courage served to bring to a head plans long talked over by the confidants of the Comte d'Artois (the future Charles X.
- Georges Cadoudal, General Pichegru and other devoted royalists had concocted with the comte d'Artois (afterwards Charles X.
- Female Desmoscolex elongatus Panceri, ventral view, X 260.
- F.,* X.)
- The tracheal system in Hexapods is very complex, 1 x, forming a series of longitudinal trunks with nexions transvers II), anastomosing finest sub-division and extendingdi by g by the After Miall and Denny, The Cock- nest su-vson and re roach, Lovell Reeve & Co.
- x, Secondary body-cavity.
- Next in order of date, though at a long interval, comes Pliny the Elder, in whose Historia Naturalis Book X.
- X.] Returning to the Old World, we have first Iceland, the fullest-indeed the only full-account of the birds of which is already stated) the extraordinary views of its adherents found little favour on the continent of Europe.
- It was accordingly regarded as the type of a distinct order Odontolcae (x.
- FELANI' X, or Felaniche, a town of Spain, in the south-east of the island of Majorca, Balearic Islands; about 5 m.
- At Z is the treasury of St Mark, which was originally one of the towers belonging to the old ducal palace; E, site of old houses; G, clocktower; H, old palace of procurators; J, old library; M, two columns; N, Ponte della Paglia; 0, Bridge of Sighs; W, Giants' Staircase; X, sacristy of St Mark; Y, Piazzetta.
- AD.; X.)
- See Strabo, pp. 401, 418, 424-425; Pausanias x.
- Its cartesian equation, when the line joining the two fixed points is the axis of x and the middle point of this line is the origin, is (x 2 + y 2)2 = 2a 2 (x 2 - y 2) and the polar equation is r 2 = 2a 2 cos 20.
- The area of the complete curve is 2a 2, and the length of any arc may be expressed in the form f(1 - x 4) - i dx, an elliptic integral sometimes termed the lemniscatic integral.
- Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .
- The elliptic lemniscate has for its equation (x 2 +31 2) 2 =a 2 x 2 +b 2 y 2 or r 2 = a 2 cos 2 9 +b 2 sin 20 (a> b).
- In the very early rock inscriptions of Thera (700-600 B.C.), written from right to left, it appears in a form resembling the ordinary Greek X; this form apparently arose from writing the Semitic symbol upside down.
- In April 1790, just after the outbreak of the French Revolution, Louis X.
- Thus if a person holds futures for 10,000 bales which stood at 5.20 on the last settlement day and now stand at 5.30, and in the course of the previous week has sold 5000 bales of " futures " at 5.1 o, he receives 10,000 X - i ce o d.
- on his old holding, and has to pay 5000 X o° 0 d.
- But in 1276 Gregory X.
- The chief sources from which the story of the Cid is to be gathered are, first, the Latin chronicle discovered by Risco in the convent of San Isidro at Leon, proved by internal evidence to have been written before 1258; the Cronica General, composed by Alphonso X.
- St Neapel, x.
- Some positive idea of his speculations may be derived from two of his observations: the one in which he notices that the parts of animals and plants are in general rounded in form, and the other dealing with the sense of hearing, which, in virtue of its limited receptivity, he compares ' If this be the proper translation of Aulus Gellius, Noctes Atticae, x.
- He prided himself on his ancient Etruscan lineage, and claimed descent from the princely house of the Cilnii, who excited the jealousy of their townsmen by their preponderating wealth and influence at Arretium in the 4th century B.C. (Livy x.
- If G is the acceleration of gravity at the equator and g that at any latitude X, then g= G(IFoï¿½o0513 sin 2 X).
- During 1822 and the succeeding years he travelled about Europe on the search for materials for his Collection des chroniques rationales fran4aises ecrites en langue vulgaire cat XIII e au X VI' siècle (47 vols., 1824-1829).
- 1); strange social upheavals may be seen: the poor 2 set in high places, the rich cast down, slaves on horseback, princes on foot (x.
- 16, 17, x.
- 5, 6, x.
- 2-4, x.
- The general historical situation, also, presupposed or referred to, is that of the period from the year 200 B.C. to the beginning of our era; in particular, the familiar references to kings as a part of the social system, and to social dislocations (servants and princes changing places, x.
- 14; x.
- In any case, since Ben-Sira belongs to about 180 B.C., the date of' Koheleth, so far as these coincidences indicate it, would not be far from 200 B.C. The contrast made in x.
- 399-575; Juvenal x.
- 1316), king of France, son of Louis X.
- 576-622; Pausanias x.
- high, approached on either side by a flight of steps leading to the top; this block, which Curtius supposes to have been the primitive altar of Zeus "T ' w ros, may be safely identified with the orators' bema, 6 X Wos Ev 7-?7 IIUKvL (Aristoph.
- The " Long Walls " (Ta µaKpet TEixn, Ta cnc X) consisted of (1) the " North Wall " (TO l36p€tov TEIXor), (a) the Walls."
- This inner and shallower harbour, perhaps the Kcw463 ?up*, was afterwards excluded from the town precinct by the walls of Conon, which traversing its opening on an embankment (76 Sta, uEuov x i.
- The arch is surmounted by a triple attic with Corinthian columns; the frieze above the keystone bears, on the north-western side, the inscription aZS' 'Aqvat, OouEw 7rpiv rats, and on the south-eastern, aZS' do' `ASptavoii Kai ou X i Ono-Los 'TO Xis.
- Omont, Athenes au X VII I siecle (Paris, 1898, with plans and views of the town and acropolis and drawings of the sculptures of the Parthenon); J.
- Pius X >>
- TRISECTRI X, a curve which is a variety of the limacon of Pascal, and named from its property of trisecting an angle.
- See The Imperial Gazetteer of India (Oxford, 1908), x.
- Thus, i part by weight of hydrogen unites with 8 parts by weight of oxygen, forming water, and with 16 or 8 X 2 parts of oxygen, forming hydrogen peroxide.
- Again, in nitrous oxide we have a compound of 8 parts by weight of oxygen and 14 of nitrogen; in nitric oxide a compound of 16 or 8 X 2 parts of oxygen and 1 4 of nitrogen; in nitrous anhydride a compound of 24 or 8 X 3 parts of oxygen and 14 of nitrogen; in nitric peroxide a compound of 3 2 or 8 X 4 parts of oxygen and 14 of nitrogen; and lastly, in nitric anhydride a compound of 4 o or 8 X 5 parts of oxygen and 14 of nitrogen.
- Di-derivatives x x x p v as $ v as s Here we have assumed the substituent groups to be alike; when they are unlike, a greater number of isomers is possible.
- Syntheses of the Benzene Ring.-The characteristic distinctions NH NH, r.-NH, [[Cooh Cooh, _ + Nh2 Nh, H., Cooh]] x x x Tri.
- derivatives Tetra-derivatives x x x x x x ?
- x u x?X x x HZ which exist between aliphatic and benzenoid compounds make the transformations of one class into the other especially interesting.
- only contain single carbon linkages, then the number of such linkages is 2n - m, and if the thermal effect of such a linkage be X, then the termAisobviously equal to (2n - m)X.
- The value of H then becomes H =na+2m#- (2n - m)X or n +mn, where and 7 7 are constants.
- Thomsen deduces the actual values of X, Y, Z to be 14.71, 13.27 and zero; the last value he considers to be in agreement with the labile equilibrium of acetylenic compounds.
- To reduce these figures to a common standard, so that the volumes shall contain equal numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' (x, ?i, w), which are such that, combined with the axial angles, they enclose volumes which contain equal numbers of molecules.
- The equivalent volumes and topic parameters are tabulated: From these figures it is obvious that the first three compounds form a morphotropic series; the equivalent volumes exhibit a regular progression; the values of x and t,t, corresponding to the a axes, are regularly increased, while the value of w, corresponding to the c axis, remains practically unchanged.
- 3.4) acid is more closely related to m-nitrobenzoic acid, x and ' being increased, w diminished.
- By taking appropriate differences the following facts will be observed: (1) the replacement of potassium by rubidium occasions an increase in the equivalent volumes by about eight units, and of rubidium by caesium by about eleven units; (2) replacement in the same order is attended by a general increase in the three topic parameters, a greater increase being met with in the replacement of rubidium by caesium; (3) the parameters x and, p are about equally increased, while the increase in w is always the greatest.
- It will be seen that (1) the increase in equivalent volume is about 6.6; (2) all the topic parameters are increased; (3) the greatest increase is effected in the parameters x and tG, which are equally lengthened.
- If the crystal structure be regarded as composed of 0 three interpenetrating point systems, one consisting of sulphur atoms, the second of four times as many oxygen atoms, and the third of twice as many potassium atoms, the systems being so arranged that the sulphur system is always centrally situated with respect to the other two, and the potassium system so that it would affect the vertical axis, then it is obvious that the replacement of potassium by an element of greater atomic weight would specially increase the length of w (corresponding to the vertical axis), and cause a smaller increase in the horizontal parameters (x and 1/ '); moreover, the increments would advance with the atomic weight of the replacing metal.
- In the topic parameters the following changes were recorded: replacement of potassium by ammonium was attended by a considerable increase in w, x and V.
- being equally, but only slightly, increased; replacement of phosphorus by arsenic was attended by a considerable increase, equally in x and >', while w suffered a smaller, but not inconsiderable, increase.
- A, Five specimens of Echinorhynchus acus, Rud., attached to a piece of intestinal wall, X 4.
- Pauthe, Bourdaloue (les maitres de la chaire au X VIIe siecle) (Paris, 1900); E.
- Some, however, see in it a corruption of the Semitic name samekh, the letter which corresponds in alphabetic position and in shape to the Greek (x).
- Further points of difficulty in connexion with the sibilants are discussed under X and Z.
- P. Mahaffy, Ale x ander's Empire (" Stories of the Nations Series "); Progress of Hellenism in Alexander's Empire (1905) The Silver Age of the Greek World (1906).
- For Cicero's opinion see Brutus, 82; Quintilian x.
- Becker, X.
- No such exaggeration exists in the case of reliefs of parts of the Alps, on a large scale, by P. Keil and Pelikan (1890), X.
- Coote's Remarkable Maps of the X Vth, X Vlth and X VIIth Centuries reproduced in their Original Size (Amsterdam, 1894-1897), and Bibliotheca lindesiana (London, 1898) with facsimiles of the Harleian and other Dieppese maps of the 16th century.
- Marcel's Reproductions de cartes et de globes relatives a la decouverte de l'Amerique du XVI' au X VIII siecle (Paris, 1893) and E.
- Through his mother he was a grandson of Louis X.
- The words ascribed to Christ in Luke x.
- ` Hv6 X aQOV µ7 7 EA p s.
- His successor was Pius X.
- I.; X.)
- A, Tapir (Tapirus indicus), X 1.
- B, Rhinoceros (Rhinoceros sumatrensis), X g.
- C, Horse (Equus caballus), X s.
- Pope Leo X.
- The second part of the book (x.-xix.) connects itself formally with the first by a summary description of the role of wisdom in the early times: she directed and preserved the fathers from Adam to Moses (x.
- On the other hand, it must be admitted that the points of view in the two parts are very different: the philosophical conception of wisdom and the general Greek colouring, so prominent in the first part, are quite lacking in the second (x.
- In the days of Jehu the country was taken from Israel by Hazael, king of Syria (2 Kings x.
- Also a unit class is any class with the property that it possesses a member x such that, if y is any member of the class, then x and y are identical.
- A doublet is any class which possesses a member x such that the modified class formed by all the other members except x is a unit class.
- If s is any class and zero is a member of it, also if when x is a cardinal number and a member of s, also x-}-I is a member of s, then the whole class of cardinal numbers is contained in s.
- A relation (R) is serial when (I) it implies diversity, so that, if x has the relation R to y, x is diverse from y; (2) it is transitive, so that if x has the relation R to y, and y to z, then x has the relation R to z; (3) it has the property of connexity, so that if x and y are things to which any things bear the relation R, or which bear the relation R to any things, then either x is identical with y, or x has the relation R to y, or y has the relation R to x.
- Two relations R and R' are said to be ordinally similar, if a one-one relation holds between the members of the two fields of R and R', such that if x and y are any two members of the field of R, such that x has the relation R to y, and if x' and y are the correlates in the field of R' of x and y, then in all such cases x has the relation R' to y', and conversely, interchanging the dashes on the letters, i.e.
- R and R', x and x', &c. It is evident that the ordinal similarity of two relations implies the cardinal similarity of their fields, but not conversely.
- If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x = m X y.
- If a is a real number, +a is defined to be the relation which any real number of the form x+a bears to the real number x, and - a is the relation which any real number x bears to the real number x+a.
- If such a complex number is written (as usual) in the form x i e l +x 2 e 2 +...
- +xnen, then this particular complex number relates x i to I, x 2 to 2,.
- The sum of two complex numbers x i e l +x 2 e 2 + ...
- +ynen is always defined to be the complex number (x i +yl)ei+(x2+y2)e2+...
- The product of two complex numbers of the second order - namely, l e l +x 2 e 2 and y i e l +y 2 e 2, is in this case defined to mean the complex (x i y i - x 2 y 2)e i +(x i y 2 +x 2 y 1)e 2.
- Thus e1Xe1 = el, e 2 Xe 2 = - e l, e i X e 2 =e 2 X e 1 =e 2.
- Call this class w; then to say that x is a w is equivalent to saying that x is not an x.
- Call it a propositional function; and, if 4)x be a propositional function, the undetermined variable x is the argument.
- Now consider a propositional function Fx in which the variable argument x is itself a propositional function.
- Accordingly, it is a fallacy for any determination of x to consider "x is an x" or "x is not an x" as having the meaning of propositions.
- Note that for any determination of x, "x is an x" and "x is not an x," are neither of them fallacies but are both meaningless, according to this theory.
- (X.) Literature.
- (Jerome Napoleon) and X.
- His orders were at once issued and complied with with such celerity that by the 31st he stood prepared to advance with the corps of Soult, Ney, Davout and Augereau, the Guard and the reserve cavalry (80,000 men on a front of 60 m.) from Myszienec through Wollenberg to Gilgenberg; whilst Lannes on his right towards Ostrolenka and Lefebvre (X.) at Thorn covered his outer flanks.
- 10, is associated in Jewish tradition with the barley harvest (Mishna, Menachoth x.).
- is an admirable exposition of the narrative contained in St Mark's Gospel x.
- The Flos of Leonardo turns on the second question set by John of Palermo, which required the solution of the cubic equation x 3 -{-2x'-}-lox = 20.
- Leonardo, making use of fractions of the sexagesimal scale, gives X = I° 221 7 42" i 33 iv 4v 40 vi, after having demonstrated, by a discussion founded on the 10th book of Euclid, that a solution by square roots is impossible.
- Perfectly pure distilled sea-water dissociates, to an infinitesimal degree, into hydrogen (H) and hydroxyl (HO) ions, so that one litre of such water contains 1 X 10 7, or 1 part of a gram-molecule of either hydr010,000,000 gen or hydroxyl (a gramme-molecule of hydrogen is 2 grammes, or of hydroxyl 17 grammes).
- Soc. (x.
- Crauford, x xl.
- (X.) Ark of the Covenant, Ark of the Revelation, Ark of the Testimony, are the full names of the sacred chest of acacia wood overlaid with gold which the Israelites took with them on their journey into Palestine.
- The other limits of the Ammonitis, or country of the Ammonites ('Aï¿½ï¿½av7res x pa, 2 Mac. iv.
- vi.; Ausonius (Gratian's tutor), especially the Gratiarum actio pro consulatu; Symmachus x.
- Taking the chemical equivalent weight of silver, as determined by chemical experiments, to be 107.92, the result described gives as the electrochemical equivalent of an ion of unit chemical equivalent the value 1 036 X 5.
- If, as is now usual, we take the equivalent weight of oxygen as our standard and call it 16, the equivalent weight of hydrogen is I o08, and its electrochemical equivalent is I 044 X 5.
- electromagnetic unit of current, this number becomes I 036 X 4.
- The ratios of the coagulative powers can thus be calculated to be i: x: x 2, and putting x =32 we get I: 32: 1024, a satisfactory agreement with the numbers observed.4 The question of the application of the dissociation theory to the case of fused salts remains.
- Let x be the number of molecules which dissociate per second when the number of undissociated molecules in unit volume is unity, then in a dilute solution where the molecules do not interfere with each other, xp is the number when the concentration is p. Recombination can only occur when two ions meet, and since the frequency with which this will happen is, in dilute solution, proportional to the square of the ionic concentration, we shall get for the number of molecules re-formed in one second ye where q is the number of dissociated molecules in one cubic centimetre.
- 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.
- The dynamical equivalent of the calorie is 4.18 X Io 7 ergs or C.G.S.
- units of work, and therefore the electromotive force of the cell should be 1.112 X Io 8 C.G.S.
- If we take as an example a concentration cell in which silver plates are placed in solutions of silver nitrate, one of which is ten times as strong as the other, this equation gives E = o 060 X Io 8 C.G.S.
- He rightly regarded the accession of Charles X.
- Frederick expressed the desire to make the personal acquaintance of his conqueror; and Charles X.
- Various church councils prohibited it, and the Code of 'Alfonso X.
- On his return home he met King Charles X.
- X 90.
- named Chytri (feast of pots, from X &rpos, a pot), a festival of the dead.
- E.; X.)
- Let k such transpositions be necessary; then the expression X(kal aa2 N a 3.
- 1 (1 +/-lD1+Fl2D2+ï¿½3D3+...) (X i X 2 X 3 ...) ï¿½ Comparing coefficients of like powers of A we obtain DX1(X1X2X3...) = (X2X3...), while D 8 (X 1 X 3 X 3 ...) =o unless the partition (X3X3X3...) contains a part s.
- In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.
- In general in space of n dimensions we have n substitutions similar to X l = a11x1 +a12x2 + ï¿½ ï¿½ ï¿½ + ainxn, and we have to express the n 2 coefficients in terms of Zn(n - I)i independent quantities; which must be possible, because X1+X2+..."IL Xn =xi+x2 +x3 +...+4.
- For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.
- Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + ï¿½ - A 1 and the adjoint is 1+A v +Aï¿½ -ï¿½ +Av -v +Aï¿½ 1+11 2 A +/-tv pt+AvA +ï¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+Aï¿½)X2 +2(/1 +Av)X3 1bx2 = 2(Aï¿½ - v)Xl+(1 +ï¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +ï¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.
- Suppose n dependent variables yl, y2,ï¿½ï¿½ï¿½yn, each of which is a function of n independent variables x1, x2 i ï¿½ï¿½ï¿½xn, so that y s = f s (x i, x 2, ...x n).
- From the differential coefficients of the y's with regard to the x's we form the functional.
- x l, x 2,...
- x n / I l yl, y2,ï¿½ï¿½ï¿½yn Theorem.-If the functions y 1, y2,ï¿½ï¿½ï¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.
- Hence if A does not vanish x 1 = x 2 =...
- We can solve these, assuming them independent, for the - i ratios yl, y2,...yn-iï¿½ Now a21A11 +a22Al2 ï¿½ ï¿½ ï¿½ = 0 a31A11+a32Al2 +ï¿½ ï¿½ï¿½ +a3nAln = 0 an1Al1+an2Al2 +ï¿½ï¿½ï¿½+annAln =0, and therefore, by comparison with the given equations, x i = pA11, where p is an arbitrary factor which remains constant as i varies.
- f(x) = f =a o xm "- a l + a 2 xm-2 - ...
- = 0, 4(x) = 4) = box' -bix'-1+.b2xn-2-...
- Now, suppose f and 4) to have a common factor x--y, f(x) =f1(x)(x--y); 4,(x) =4,1(x)(x--y), f l and 41 being of degrees m-1 and ni respectively; we have the identity ch i (x)f(x) =fl(x)4,(x) of degree m+n-I.
- Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.
- yl, y 2,...yn) (zl, z2,...zn z1, z 2, ï¿½ï¿½ï¿½zn xi, 'X' 2,...
- He forms n equations from f by separate multiplication by x, ...x, I, in succession, and similarly treats 4) with m multipliers I.
- He forms the equation .f()4(') -.f(x')4)(x) = o, which can be satisfied when f and 4 possess a common factor.
- He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m.
- Taking two of the equations ax + +cz) x"' 1 +...
- = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.
- If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.
- 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 = ...
- af Expression in Terms of Roots.-Since x+y y =mf, if we take cx any root x 3, y1, ofand substitute in mf we must obtain, y 1 C) zaZ1 ï¿½; hence the resultant of and f is, disregarding numerical factors, y,y2...y,,.
- 1 X discriminant of f = ao X disct.
- Now f = (xy 1 - x i y) (xy 2 - x 2 y) ...
- (x y m - x m y), ar _ y1(x y 2 - and substituting in the latter any root of f and forming the product, we find the resultant of f and d, viz.
- y m (xly2 - x2y1) 2 (x0,3 - x 3 yl) 2...
- (x rys - x8yr) 2...
- If we write (I +a i x) (I a 2 x) ...
- (I x n x) = I +a l x+ a 2 x 2 -{-...
- 1 +hlx-+h2x2+h3x3-}-..., which remains true when the symbols a and h are interchanged, as is at once evident by writing -x for x.
- Multiplying out the right-hand side and comparing coefficients X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being for all partitions of m.
- on the right-hand side is such that the coefficient of x n ix n Zx n 3...
- in 1 "1142 P3 X ?
- ) j1+j2+j3+..ï¿½ (J1+ j2 +j3+...-1)!/T1)?1(J2)72 (J 3)/3..., j11j2!j3!... ?.1 for the expression of Za n in terms of products of symmetric functions symbolized by separations of (n 1 1n 2 2n 3 3) Let (n) a, (n) x, (n) X denote the sums of the n th powers of quantities whose elementary symmetric functions are a l, a 2, a31ï¿½ï¿½ï¿½; x 1, x2, x31..; X1, X2, X3,...
- respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3...
- in terms of x 1, x2, x3,ï¿½ï¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If ï¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ...
- X,1X82>$3...=...+8(m m m ...)x 11 x 12 x13......
- 1 2 3 We have found above that the coefficient of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ...
- Putting x1= I and x 2 = x 3 = x 4 = ...
- _ ...+o(mi and another by putting x i = x 2 = x3= ...'
- =I, for then X.
- be any partitions of X, respectively, the function isexpressible by means of functions symbolized by separation of X1A2X 3.
- ï¿½ ï¿½ P1 v2 v3...) ï¿½ For, writing as before, Xm 'Xm 2 Xm '= zzo(SQls:2s73...) xi'x12x13..., 1 2 3" 1231 2 3 = EPxi l x A2 x A3, P is a linear function of separations of(/ 1 / 2 A2 / 4 3 3 ...) of specification (m"`1mï¿½2m"`3...), and if X; 1 X 3 2X8 3 ' ..
- = EP x tl x t2 x t3 ...
- - Starting with the relation (1 + a i x) (1 +a 2 x)...
- (1 +a n x) = 1 +a 1 x+a 2 x 2 +...
- +aï¿½xn multiply each side by I +px, thus introducing a new quantity A; we obtain (1 +a1x) (1+a2x)...(1 -Fanx)(1+,ux) = 1+(a1 +1a)x + (a2+1aa1)x2+...
- The introduction of the quantity p converts the symmetric function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.ï¿½.) +/103(A1X2.ï¿½.) +....
- -} p3D8 ...) fl X (1 +/lDl+ï¿½2D2+...+Asps+...) f2 X.
- X (1 +PD1+12D2+...+ï¿½8D8+...) fm, and now expanding and equating coefficients of like powers of /t D 1 f - Z(Difi)f2f3.
- is transformed into the operator d 1 by the substitution (ac, al, a2, ï¿½ï¿½ï¿½as, ï¿½ï¿½ï¿½) _ (ao, Xoai, X 6 X i a 2, ï¿½ï¿½ï¿½, XcX1..%s_las,ï¿½ï¿½ï¿½), so that the theory of the general operator is coincident with that of the particular operator d1.
- For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,...
- =0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-2 -...
- Further, let 1 -1-b i x+ b 2 x 2' +...
- +bmx m = (1 +Q 1 x) (1 +0 2 x)...
- If m be infinite and 1 + b i x + b 2 x 2 +...
- (1 + a i x) (1+ = s i z we have the symbolic identity +02712+0.3x3+...
- In this notation the fundamental relation is written (l + a i x +01Y) (I + a 2x+l32Y) (1 + a3x+133y)...
- = 1 +(l A x +(01) y +(102) x2 +(1001)xy+(512)3,2 +(103)x 3 +(10201)x i y+(10 O12)xy2+ (013)y3+...
- For write (pq) =sï¿½ and take logarithms of both sides of the fundamental relation; we obtain slox +soot' = + (3ly) S20x 2 +2siixy+s02y 2 = E(aix+(3 ly) 2, &C., and siox+SOly - (S 20 x2 + 2s ii x y+ s ooy 2) +...
- = exp {(siox+Solt') - s20 x 2+ 2siixy+S02y2)+ï¿½ï¿½ï¿½}, and thence derive the formula?
- - If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -ï¿½-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al °, aol, ...) =f exp(ï¿½dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem.
- It may be written in the form n n-1 2 ax 1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ?
- Other forms are n-1 n-2 2 ax +nbx x +n(n-i)cx x +..., 1121 2 the binomial coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2 ax 1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later.
- For present purposes the form will be written a0x 1 +(7)a1x1=1 x2+ C 2)o'2x12 x 2 +...+anx2, the notation adopted by German writers; the literal coefficients have a rule placed over them to distinguish them from umbral coefficients which are introduced almost at once.
- If the variables of the quantic f(x i, x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +ï¿½ï¿½ï¿½ in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.
- By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.
- x i, x 2) is said to be a covariant of the quantic. The expression " invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with.
- Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...
- Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.
- 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?
- For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2.
- If we solve the equations connecting the original and transformed unbrae we find (A ï¿½) (- a 2) =A i( - A 2) + ï¿½'1A1, (A ï¿½) a1 = A2(- A2)+ï¿½2A1, and we find that, except for the factor (A /), -a 2 and +ai are trans formed to -A 2 and +A i by the same substitutions as x i and x 2 are transformed to i and E2.
- For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2.
- We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:-(i) " If two equally numerous sets of quantities x, y, z,...
- x', y', z',...
- are such that whenever one set x, y, z,...
- is expressed in terms of new quantities X, Y, Z, ...
- the second set x', y', z', ...
- is expressed in terms of other new quantities X', Y', Z', ....
- (2) " Two sets of quantities x, y, z, ...; E, n, i, ...
- are said to be contragredient when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z ï¿½..., Z = A 3 X +ï¿½3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?.
- are contragredient with the d- variables x, y, z, ...
- for when (x, z, ï¿½ï¿½ï¿½) = (A l, ï¿½i, VI I ï¿½ï¿½ï¿½) (X, Y, Z, ï¿½ï¿½ï¿½), I A 2, / 2 2, Y2, ...
- If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...
- the i t " power of that appertaining to a x and b x multiplied by the j t " power of that appertaining to a x and c x multiplied by &c. If any two of the linear forms, say p x, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero.
- may be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, ....
- are equal to a x, x, c x, ...
- respectively, the linear forms a x, b., cg, ...
- In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ?
- For, if c(ao i ...x l, x 2) be a covariant of order e appertaining to a quantic of order n, t (T.
- From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0.
- We may in any relation substitute for any pair of quantities any other cogredient pair so that writing -}-d 2, -d l for x 1 and x 2, and noting that gx then becomes (gd), the above-written identity bceomes (ad)(bc)+(bd)(ca)+(cd)(ab) = 0.
- (IV.) and herein writing d 2, -d 1 for x l, x2, 2 (ac) (bc) (ad) (bd) = (bc) 2 (ad) 2 +(ac) 2 (bd) 2 - (ab) 2 (cd) 2.
- 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+...
- -2 _ ab 2an-2bn-2Crz z x () x x x, Each term on the right-hand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant; they are equivalent symbolic products, and we may accordingly write 2(ac) (bc)ai -1 bi -1 cx 2 =(ab)2a:-2b:-2c:, a relation which shows that the form on the left is the product of the two covariants n (ab) ay 2 by 2 and cZ.
- (ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant.
- of the symbolic factors of the form are replaced by IA others in which new variables y1, y2 replace the old variables x1, x 2 .
- - We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b.
- (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.
- 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.
- Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, ï¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.
- It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2ï¿½ It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.
- - An important method for the formation of covariants is connected with the form f +X4), where f and 4 are of the same order in the variables and X is an arbitrary constant.
- If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4).
- X1, X 2, u1, /22 being as usual the coefficients of substitution, let x1a ?
- + X 2 - = D, X 1 -' j +X 2 =D 2 AA' ?2 / 2 1 3 - 5 -, =112 87,2 = ?1a a + ?2a a =Dï¿½ï¿½, 1 be linear operators.
- The first and fourth of these indicate that (a 2) w is a homogeneous function of X i, X2, and of /u1, ï¿½ 2 separately, and the second and third arise from the fact that (X / 1) is caused to vanish by both Da ï¿½ and Dï¿½A.
- Now 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-+...
- If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.
- In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.
- From (ac) 2 (bd) 2 (ad)(bc) we obtain (bd) 2 (bc) cyd x +(ac) 2 (ad) c xdx - (bd) 2 (ad)axb x - (ac)2(bc)axbx =4(bd) 2 (bc)c 2.
- d x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic.
- The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases.
- Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.
- First observe that with f x =a: = b z = ï¿½ï¿½ï¿½,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.
- and that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x.
- % -k Y k = (af) k a n x.
- ukx(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?
- Moreover 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.
- To exhibit any covariant as a function of uo, ul, a n = (aiy1+a2y2) n and transform it by the substitution fi y 1+f2 y where f l = aay 1, f2 = a2ay -1, x y - x y = X x thence f .
- y1 = x 15+f2n; fï¿½ y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jnï¿½ 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.
- The first transvectant, (f,f') 1 = (ab) a x b x, vanishes identically.
- the two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f,, >) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 .
- If (f,4) 1 be not a perfect square, and rx, s x be its linear factors, it is possible to express f and 4, in the canonical forms Xi(rx)2+X2(sx)2, 111(rx)2+1.2 (sx) 2 respectively.
- In fact, if f and 4, have these forms, it is easy to verify that (f, 4,)i= (A j z) (rs)r x s x .
- The Hessian 0 =A 2 is such that (f, 2 and if f is expressible in the form X(p x) 3 +,i(g x) 3, that is as the sum of two perfect cubes,.
- we find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +ï¿½(g x) 3 and immediately solved.
- The simplest form to which the quartic is in general reducible is +6mxix2+x2, involving one parameter m; then Ox = 2m (xi +x2) +2 (1-3m2) x2 ix2; i = 2 (t +3m2); j= '6m (1 - m) 2; t= (1 - 9m 2) (xi - x2) (x21 + x2) x i x 2.
- The .sextic covariant t is seen to be factorizable into three quadratic factors 4 = x 1 x 2, =x 2 1 - 1 - 2 2, 4) - x, which are such that the three mutual second transvectants vanish identically; they are for this reason termed conjugate quadratic factors.
- of 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.
- We 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, ?
- If 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.
- The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (Xï¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence?
- T = (j, j) 2 jxjx; 0 = (iT)i x r x; four other linear covariants, viz.
- a = - (ji) 2 jx; s = (ia)ix; Y = (ra)r x: (3= (T0)T x .
- When C vanishes j has the form j = pxg x, and (f,j) 3 = (ap) 2 (aq)ax = o.
- Hence, from the identity ax (pq) = px (aq) -qx (ap), we obtain (pet' = (aq) 5px - 5 (ap) (aq) 4 pxg x - (ap) 5 gi, the required canonical form.
- f= ai; the Hessian H = (ab) 2 azbx; the quartic i= (ab) 4 axb 2 x; the covariants 1= (ai) 4 ay; T = (ab)2(cb)aybyci; and the invariants A = (ab) 6; B = (ii') 4 .
- x x x To form an invariant or covariant we have merely to form a product of factors of two kinds, viz.
- For example, take the ternary quadratic (aixl+a2x2+a3x3) 2 =a2x, or in real form axi +bx2+cx3+2fx 2 x 3+ 2gx 3 x 1 +2hx i x 2.
- We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.
- The Hessian is symbolically (abc) 2 azbzcz = H 3, and for the canonical form (1 +2m 3)xyz-m 2 (x 3 +y 3 +z 3).
- By the x process of Aronhold we can form the invariant S for the cubic ay+XH:, and then the coefficient of X is the second invariant T.
- This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the number of ground forms is apparently very great.
- He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
- When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.
- When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.
- l 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 -?
- Let a covariant of degree e in the variables, and of degree 8 in the coefficients (the weight of the leading coefficient being w and n8-2w = ï¿½), be Coxl -}- ec l l 1 x 2 -{-...
- a 31 ...n !aï¿½, for a l, a 2, a 3 ...a n respectively; it then becomes a e xi +na l xi -I x 2 +n (n -1)a 9 xl -2 x2 +...
- from the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.
- Similarly, if 0 =3, every form (3K+12,x) is a perpetuant.
- Forms.-Taking the two forms to be a o xi + pa l x i 1x2+p(p-1)a2xr2x2-I-...
- +Aexe .(1 +Te'x) =1 +Bix+B2x2+...+Bo'xe' Al+B1=0.
- 1-e may be represented by the form (1 X 1 +1) a (2 g 2 +1) b - (1A1)a(2g2+11)b+ ...
- t (22P11),; X 1 and 12 each assuming all integer (including zero) values.
- (1 + r i x)(1 where For the case 0=i, 0' =1, apparently, a choice of four products.
- a 0(3 ï¿½3+l 2 ï¿½2+1 1 Ai+2)b, For the case 0=2, 0' =3, the condition is a102T1T2T3(01+ 172)(61 +T1)(171+T2)(1T1+T3)(a2+T1)(12+T2)(cT2+T3) X (T 1 +T2) 3 (T 2 +T 3) = 0.
- Taking the variables to be x, y and effecting the linear transformation x = X1X+1.11Y, y = X2X+It2Y, X 2 +Y2X Y Xl - X2 y = _ x X I + AI R X 122 so that - ï¿½l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.
- As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes.
- The substitution for x, y in terms of X, Y is the most general linear substitution in virtue of the four degrees of arbitrariness introduced, viz.
- Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.
- 2 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.
- In general the Boolian system, of the general n i °, is coincident with the simultaneous system of the n i °' and the quadratic x 2 +2 cos w xy+y2.
- We have cos w' = cos w = o and the substitution x 1 =cos OX, -sin 0(2 x 2 = sin OX i +cos 6X2, with modulus unity.
- This is called the direct orthogonal substitution, because the sense of rotation from the axis of X i to the axis of X, is the same as that from that of x i to that of x 2.
- If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.
- In both cases ddl and dal are cogredient with xl and x 2; for, in the case of direct substitution, dxi = cost dX i - sin 00-(2, ad2 =sin B dX i +cos O dX 2, and for skew substitution dai = cos B dX i +sin 0d2, c-&-- 2 n d =sin -coseax2.
- In the a x = aixi+a2x2, observe that a a = a2, ab = aibi +a2b2.
- then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - ï¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.
- Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.
- X (xa) ki (xb) k2 (xc) k3...axibx2cx3...xx = (AB) hi (AC) h2 (BC) h3...A11 4 13 A1,14131 A B I ?C"' B C "' X (XA) ki (XB) k2 (XC) k3...AXB122cCk...X If this be of order e and appertain to an nie L Eke-/1+2m =e, h i+h2+ï¿½ï¿½ï¿½+221+ji+j2+ï¿½ï¿½ï¿½+kl+li =n, hi+h3+..ï¿½+222+ji+j3+ï¿½ï¿½ï¿½+k2+12 = n, h2+h3+ï¿½ï¿½ï¿½+223+j2+%3+ï¿½.ï¿½+k3+13 =n; viz., the symbols a, b, c,...
- or to a power of x .