# X sentence example

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.
- 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.
- 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.
- He published Lives of Foreign Statesmen (1830), The Greek and the Turk (1853), and Reigns of Louis X VIII.Advertisement
- 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.
- 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.
- This page gives an overview of all articles in the 1911 Brittanica which are alphabetized under X to Yve.Advertisement
- 1 These ceased to have legal currency at the end of igoi; they were notes of x and 2 lire.
- Aulard, Les Portraits litteraires a la fin du X VIII" siecle, pendant la Revolution (Paris, 1883).
- 2 Considerations sur les corps organises, chap. x.
- 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.
- - End view of skull of a Chicken fo three weeks old, X 8 diameters.Advertisement
- 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.
- 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.
- 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.
- 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.Advertisement
- 6 According to the most trustworthy accounts, but see Letters and Papers, x.
- 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.
- 'Invous X ptar6s, Oeou `Tuffs, 16 y TIJp, Jesus Christ, Son of God, Saviour, which together spell the Greek word for "fish," ix9vs.
- - X I I I, XIV, thirteenth the intestine.Advertisement
- 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.Advertisement
- 3, Cups enlarged X 5.
- 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.
- 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.
- 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.Advertisement
- 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.
- 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.Advertisement
- 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.
- 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.
- 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.Advertisement
- The " Long Walls " (Ta µaKpet TEixn, Ta cnc X) consisted of (1) the " North Wall " (TO l36p€tov TEIXor), (a) the Walls."
- 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.
- 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.Advertisement
- 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.
- 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.
- 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.
- A, Five specimens of Echinorhynchus acus, Rud., attached to a piece of intestinal wall, X 4.
- 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).
- 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.
- The words ascribed to Christ in Luke x.
- ` Hv6 X aQOV µ7 7 EA p s.
- 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.
- 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.
- 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.
- 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.).
- 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).
- (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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- = 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 = ...
- Now f = (xy 1 - x i y) (xy 2 - x 2 y) ...
- If we write (I +a i x) (I a 2 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.
- 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 = ...
- - 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.) +....
- X (1 +PD1+12D2+...+ï¿½8D8+...) fm, and now expanding and equating coefficients of like powers of /t D 1 f - Z(Difi)f2f3.
- 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' +...
- If m be infinite and 1 + b i x + b 2 x 2 +...
- In this notation the fundamental relation is written (l + a i x +01Y) (I + a 2x+l32Y) (1 + a3x+133y)...
- 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+ ?
- 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.
- 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,...
- 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, ...
- In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ?
- 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.
- (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.
- (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.
- 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.
- 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.
- 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.
- 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 .
- The first transvectant, (f,f') 1 = (ab) a x b x, vanishes identically.
- 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 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 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.
- When C vanishes j has the form j = pxg x, and (f,j) 3 = (ap) 2 (aq)ax = o.
- 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).
- 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.
- 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 -{-...
- 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+ ...
- 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.
- 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 the a x = aixi+a2x2, observe that a a = a2, ab = aibi +a2b2.
- 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,...
- To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (ax) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan.
- Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through one of the circular points at infinity.
- In general any pencil of lines, connected with the line a x by descriptive or metrical properties, has for its equation a rational integral function of the four forms equated to zero.
- There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .
- If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipotential surfaces.
- 3) by F, and its components parallel to the co-ordinate axes by X and Y, we have X= - ax = M(3 cos' 0 - I), Y= - y = M (3 sin 0 cos 0.
- If F T is the force along r and F t that along t at right angles to r, F r =X cos 0+ Y sin 0=M 2 cos 0, F t = - X sin 0+ Y cos 0 = - r 3 sin 0.