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binary

A binary form which has a square factor has its discriminant equal to zero.

8076The nomenclature of acids follows the same general lines as that for binary compounds.

3029BINARY SYSTEM, in astronomy, a system composed of two stars revolving around each other under the influence of their mutual attraction.

3033BINARY SYSTEM, in astronomy, a system composed of two stars revolving around each other under the influence of their mutual attraction.

2734- An important class of invariants, of several binary forms of the same order, was discovered by Sylvester.

2229Thus, the affinity of hydrogen and oxygen for each other is extremely powerful, much heat being developed by the combination of these two elements; when binary compounds of oxygen are decomposed by the electric current, the oxygen invariably appears at the positive pole, being negative to all other elements, but the hydrogen of hydrogen compounds is always disengaged at the negative pole.

1922Thus, the affinity of hydrogen and oxygen for each other is extremely powerful, much heat being developed by the combination of these two elements; when binary compounds of oxygen are decomposed by the electric current, the oxygen invariably appears at the positive pole, being negative to all other elements, but the hydrogen of hydrogen compounds is always disengaged at the negative pole.

1821A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

1821-When the binary form a y = (alxl +a2x2)n is transformed to A;.

1827The binary products e i e j, however, are expressible as linear functions of the units e i by means of a " multiplication table " which defines the special characteristics of the algebra in question.

1722This view was accepted in 1817 by Leopold Gmelin, who, in his Handbuch der Chemie, regarded inorganic compounds as being of binary composition (the simplest being oxides, both acid and basic, which by combination form salts also of binary form), and organic compounds as ternary, i.e.

1724As regards simultaneous binary forms, the system of two quadratics, and of any number of quadratics, is alluded to above and has long been known.

1724The binary products e i e j, however, are expressible as linear functions of the units e i by means of a " multiplication table " which defines the special characteristics of the algebra in question.

1622So the theory of the forms appertaining to a binary form of unrestricted order was first worked out by Cayley and P. A.

1419If f =ay, 4 = b' be any two binary forms, we generalize by forming the function (m-k)!

1419Gordan has also shown that the vanishing of the Hessian of the binary n ic is the necessary and sufficient condition to ensure the form being a perfect n th power.

1419a V(I - a) This constant k gives a numerical value for the chemical affinity, and the equation should represent the effect of dilution on the molecular conductivity of binary electrolytes.

1420Gordan has also shown that the vanishing of the Hessian of the binary n ic is the necessary and sufficient condition to ensure the form being a perfect n th power.

1420Points on the side AB will correspond to binary alloys containing only A and B, and so on.

1215The Binary Sextic.-The complete system consists of 26 forms, of which the simplest are x2y2z2 + (1 +8 m3) 2 (y3z3 +z3x3 +x3y3).

1216By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, - 'el ' 'e2 ' 'e3 we may construct derived units of the third, fourth ...

1216We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.

1217Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are symbolical respreentations of the real coefficients ï¿½o, ai,...

1014One advantage we have obtained is that, if we now write ao =o, and substitute a 8 _ 1 for a,, when s>o, we obtain d d aO da l +al da 2 +a2 da ï¿½....+an_2dan_1 which is the form of SZ for a binary (n- Henceby merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant of the same degree, and of weight w-8, appertaining to the (n-I) ic. Also, if we increase each suffix in a seminvariant, we obtain terms, free from a 0, of some seminvariant of degree 8 and weight w+8.

1015In the Lavoisierian nomenclature acids were regarded as binary oxygenated compounds, the associated water being relegated to the position of a mere solvent.

913A distinction was formerly made between double stars of which the components were in revolution around each other, and those in which no relative motion was observed; but it is now considered that all double stars must really be binary systems.

914Consider two binary equations of orders m and n respectively expressed' in non-homogeneous form, viz.

916Consider two binary equations of orders m and n respectively expressed' in non-homogeneous form, viz.

916-, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.

8111 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.

813Ursae majoris is a beautiful binary star, its components having magnitudes 4 and 5; this star was one of the first to be recognized as a binary - i.e.

813To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a symbolic form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 ï¿½ al = a;b4 -}-alb?

814Associated Forms.-A system of forms, such that every form appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to obtain.

814Associated Forms.-A system of forms, such that every form appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to obtain.

814The process is not applicable with complete success to quintic and higher ordered binary forms. This arises from the circumstance that the simple syzygies between the ground forms are not all independent, but are connected by second syzygies, and these again by third syzygies, and so on; this introduces new difficulties which have not been completely overcome.

710But the same relation does not hold of a satellite the mass of whose primary is not regarded as an absolutely known quantity, or of a binary star.

69Before discussing these it is best to trans form the binary form by substituting I !a i, 2 !

69The Binary Cubic.-The complete system consists of f=aa,(f,f')'=(ab)2a b =0 2, (f 0)= (ab) 2 (ca)b c=Q3, x x x x x x and (0,0')2 (ab) 2 (cd) 2 (ad) (bc) = R.

611The Binary Cubic.-The complete system consists of f=aa,(f,f')'=(ab)2a b =0 2, (f 0)= (ab) 2 (ca)b c=Q3, x x x x x x and (0,0')2 (ab) 2 (cd) 2 (ad) (bc) = R.

611These may be written, for the binary nie, Zka k _.

612Salts formed from hydracids terminate in -ide, following the rule for binary compounds.

511sin/3 + sin Consider the binary n Ee.

49The points E, E', E" are the eutectics of binary alloys.

49These facilities, coupled with the wide and fascinating field of research opened up by Sir William Herschel's discovery of the binary character of double stars, gave an impulse to micrometric research which has continued unabated to the present time.

410If a compound contains two atoms it is termed a binary compound, if three a ternary, if four a quaternary, and so on.

410A Similar Theorem Holds In The Case Of Any Number Of Binary Forms, The Mixed Seminvariants Being Derived From The Jacobians Of The Several Pairs Of Forms. If The Seminvariant Be Of Degree 0, 0' In The Coefficients, The Forms Of Orders P, Q Respectively, And The Weight W, The Degree Of The Covariant In The Variables Will Be P0 Qo' 2W =E, An Easy Generalization Of The Theorem Connected With A Single Form.

410For a further discussion of the binary sextic see Gordan, loc. cit., Clebsch, loc. cit.

411When binary compounds, or compounds of two elements, are decomposed by an electric current, the two elements make their appearance at opposite poles.

412THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables.

39Single binary forms of higher and finite order have not been studied with complete success, but the system of the binary form of infinite order has been completely determined by Sylvester, Cayley, MacMahon and Stroh, each of whom contributed to the theory.

39Perpetuants.-Many difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the (p1p2p3...) to where form of infinite order.

39Proceeding as we did in the case of the single binary form we find that for a given total degree 0+0', the condition which expresses reducibility is of total degree in the coefficients a and T; combining this with the knowledge of the generating function of asyzygetic forms of degrees 0, 0', we find that the perpetuants, of these degrees are enumerated by z26"'-11 -z.

39Restricted Substitutions We may regard the factors of a binary n ip equated to zero as denoting n straight lines through the origin, the co-ordinates being Cartesian and the axes inclined at any angle.

39Then a binary n", equated to zero, represents n straight lines through the origin, and the x, y of any line through the origin are given constant multiples of the sines of the angles which that line makes with two fixed lines, the axes of co-ordinates.

39All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

39Interesting stars are: a Aurigae or Capella (the goat), one of the brightest stars in the heavens, determined by Newall and Campbell to be a spectroscopic binary; [3 Aurigae, a star of the second magnitude also a spectroscopic binary; e Aurigae, an irregularly variable star; and Nova Aurigae, a "new" star discovered by Anderson in 1892, and afterwards found on a photographic plate exposed at Harvard in December 1891.

39The Binary Quadratic.-The complete system consists of the form itself, ax, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f') 2 = (ab) 2; or, in real coefficients, 2(a 0 a 2 a 2 1).

39Proceeding as we did in the case of the single binary form we find that for a given total degree 0+0', the condition which expresses reducibility is of total degree in the coefficients a and T; combining this with the knowledge of the generating function of asyzygetic forms of degrees 0, 0', we find that the perpetuants, of these degrees are enumerated by z26"'-11 -z.

39sin/3 + sin Consider the binary n Ee.

39All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

39The points E, E', E" are the eutectics of binary alloys.

39By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1ï¿½1 +a2m2) = aa a ï¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alï¿½1+a2ï¿½2) = a A ï¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a ï¿½ respectively.

310Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.

310The Binary Quadratic.-The complete system consists of the form itself, ax, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f') 2 = (ab) 2; or, in real coefficients, 2(a 0 a 2 a 2 1).

310The Binary Quartic.-The fundamental system consists of five forms ax=f; (f,f')2=(ab) 2axbx=Ax; (f,f')4=(ab) 4= 2; (f, 0)1= (ao) azsi = (ab) 2 (cb) a:b x c5 =1; (f 4) 4 = (as) 4 = (6) 2 00 2 (ca) 2 = j, viz.

310Previous to Chevreul's researches on the fats (1811-1823) it was believed that soap consisted simply of a binary compound of fat and alkali.

27The whole theory of invariants of a binary form depends upon the solutions of the equation SZ=o.

2731-52, 1 391 5 2, 45 6); and, in 1888, that of the binary septimic, which proved to be much more complicated (Math.

28Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

28The binary conception of compounds held by Berzelius received apparent support from the observations of Gay Lussac, in 1815, on the vapour densities of alcohol and ether, which pointed to the conclusion that these substances consisted of one molecule of water and one and two of ethylene respectively; and from Pierre Jean Robiquet and Jean Jacques Colin, showing, in 1816, that ethyl chloride (hydrochloric ether) could be regarded as a compound of ethylene and hydrochloric acid.

29A quick glance at the pre-built binary really caught my eye.

10Where there is a division into sixteen parts, a binary scale may be formed by dividing into groups of two, four or eight.

11Hence for a binary electrolyte the diffusion constant is measured by 2RT/F or 2UVRT/(U+V).

12The union of any two of the elements gave rise to binary compounds, such as oxides, acids, sulphides, &c. A substance containing three elements was a binary compound of the second order; thus salts, the most important compounds of this class, were formed by the union of acids and oxides, iron sulphate, for instance, being a compound of iron oxide with sulphuric acid.

12a, Ordinary forms of varying sizes; b c, stages in multiple division; d, binary fission; e, f, g, flagellate forms (after Rogers).

13Binary system >>

13The British method is a mixture of the last two, but with an index-scale which is partly ternary and partly binary.

00OXIDE, in chemistry, a binary compound of oxygen and other elements.

00accretion process fundamental to binary stars.

00Nevertheless, despite the warning, in this event, the ace interface function passes the value to the ACE binary.

00In ternary alloys (three elements) three pairs of binary (two elements) alloys can also form.

00The destination of the gas depends on its specific angular momentum relative to the binary.

00The rules for binary arithmetic are shown in the table below.

00ASCII mode; images should be transferred in binary mode.

00If you use this installation of GAP on different hardware platforms you will have to compile the binary for each platform separately.

00In most cases it is best to specify the global cgi-bin directory as the place to install the binary.

00Alt_Name A frequently used alternative name of the cataclysmic binary.

00If the executable binary still does not run you may need to proceed with next step.

00Nevertheless, despite the warning, in this event, the ace interface function passes the value to the ACE binary.

00binary digits to a series of tones, which is the only way to pass a signal down the PSTN.

00binary compatibility has playing only a tiny role in the overall Unix picture.

00binary fission.

00binary oppositions can be traced even in visual images.

00binary pulsars (Lyne et al.

00binary arithmetic to keep each row even in value.

00BII The Galactic Latitude of the low-mass X-ray binary.

00The reasons for this lie in the rough conceptual division of the signal from a coalescing neutron star binary into two parts.

00Some rpm binary installs insist on putting programs in /opt (such as the KDE desktop (http://www.kde.org) and cannot be redirected elsewhere.

00In favorable conditions the cell divides in two by a process called binary fission.

00JOIN clauses only make sense for binary operators that return Boolean.

00There are several examples of macropolyhedral binary boranes, such as syn - and anti -B 18 H 22.

00Students will understand how a binary star system's orbit can cause changes in the observed brightness of the system.

00coalesceasons for this lie in the rough conceptual division of the signal from a coalescing neutron star binary into two parts.

00logical And Binary " and " returns the logical conjunction of the two surrounding expressions.

00The cases where the main connective in is a binary connectives are also straightforward.

00The modem converts the binary digits to a series of tones, which is the only way to pass a signal down the PSTN.

00And surely one binary digit sounds the same as another doesn't it?

00In a binary system, the white dwarf will pull material off the surface of the normal star.

00eclipseggested that Algol is what is now known as an eclipsing binary.

00encoded in a special-purpose binary format.

00The benefits of these techniques are so enormous that simple binary trees aren't used in big programs - see the appendix.

00Equality Operators Binary " == " returns true if the left argument is numerically equality Operators Binary " == " returns true if the left argument is numerically equal to the right argument.

00exists for backward binary compatibility with earlier versions of the Swing library.

00In fact they result from a thermonuclear explosion on the surface of a white dwarf star in a close binary system.

00Binary liquid explosives are a sexy staple of Hollywood thrillers.

00arithmetic expressions Operands are linked by operators: + - / * ^ which can be binary or unary.

00These grow by binary fission or budding, creating new individuals from the parent cell.

00Decimal Arithmetic Most computers today support binary floating-point in hardware.

00HALIDES Binary compounds containing anions of the elements fluorine, chlorine, bromine, iodine or astatine.

00It divides by binary fission, is Gram-positive, and strongly 'acid fast' following staining with basic fuchsin, which stains the bacteria pink.

00Similar sets of binary combinations have also been used in traditional African divination systems such as IfÃ¡ as well as in medieval western geomancy.

00To convert hex to binary, change each hex digit into four bits, using the table above.

00You can also experiment with other numbering systems, including hexadecimal and binary.

00In the binary sequence hexagram 09 opposes hexagram 16.

00The Binary Watch is a perfect accessory for the technology conscious or for the fashion trendy hipster.

00Step 2: Create science housekeeping binary file The build_hb utility is used to perform this function.

00The upper limit for the binary boron hydrides is the twelve vertex B 12 H 12 2- anion that displays perfect icosahedral geometry.

00A unary and binary (dyadic) operator giving 8 bit indirection.

00Stores the unsigned long l to the output stream in binary. virtual RWvostream& operator (short s ); Redefined from class RWvostream.

00Stores the unsigned lass="ex">unsigned long l to the output stream in binary. virtual RWvostream& operator (short s ); Redefined from class RWvostream.

00mainframe host using the " rcp -b " binary switch.

00The rightmost 6 bytes represent the mantissa in binary notation, with the leftmost bit representing the digit in the 2 -1 place.

00mantissa in binary notation, with the leftmost bit representing the digit in the 2 -1 place.

00Make sure that you download the file with your ftp client in binary mode.

00Old novae are therefore surrounded by shells of ejected matter illuminated by the light from the central binary system.

00All numbers can be expressed in 6 formats: decimal, octal, hexadecimal, binary, with fixed point, or scientific.

00However, binary oppositions occasionally masquerade the complexity of certain phenomena.

00A binary system forms first from an initial m=2 density perturbation in the cloud.

00phagocytosis of bacteria and divides by binary fission.

00The optional third argument must be a binary predicate, a binary function returning a boolean value.

00prototype implementation of a decision procedure based on Binary Decision Diagrams.

00We will have more to say about this binary pulsar later in the course.

00The binary X-ray pulsar Cen X-3 was observed over an orbital eclipse, and time variation of the iron line structure was studied.

00I compare every binary quantic with a chemical atom.. .

00self-pity form a binary, and so can switch between each other.

00self-pity mode changes to its binary, vanity.

00For synchronization, simple locks (a.k.a. mutexes or binary semaphores) are provided.

00specifyLoad binary file Format: LO filename The load binary file command loads the file specified by filename into memory.

00structuralist analysis of texts tends to focus on binary oppositions.

00This only shows the need to use more subdivisions instead binary oppositions.

00ternary alloys (three elements) three pairs of binary (two elements) alloys can also form.

00thermonuclear explosion on the surface of a white dwarf star in a close binary system.

00The change from a binary compound, that is, one containing two elements, to a ternary compound in which.

00The atomic theory can picture the change from the binary to the ternary compound simply as the addition of one or more atoms of the third element to the previously existing molecule; in such a case the combining ratio of the first two elements should be absolutely the same in both compounds.

00These facilities, coupled with the wide and fascinating field of research opened up by Sir William Herschel's discovery of the binary character of double stars, gave an impulse to micrometric research which has continued unabated to the present time.

00A distinction was formerly made between double stars of which the components were in revolution around each other, and those in which no relative motion was observed; but it is now considered that all double stars must really be binary systems.

00Interesting objects in this constellation are: a Geminorum or Castor, a very fine double star of magnitudes 2.0 and 2.8, the fainter component is a spectroscopic binary; i Geminorum, a long period (231 days) variable, the extreme range in magnitude being 3.2 to 4; Geminorum, a short period variable, 10.15 days, the extreme range in magnitude being 3.7 to 4.5; Nova Geminorum, a "new" star discovered in 1903 by H.

00Previous to Chevreul's researches on the fats (1811-1823) it was believed that soap consisted simply of a binary compound of fat and alkali.

00But the same relation does not hold of a satellite the mass of whose primary is not regarded as an absolutely known quantity, or of a binary star.

00He laid down the following arbitrary rules for determining the number of atoms in a compound: - if only one compound of two elements exists, it is a binary compound and its atom is composed of one atom of each element; if two compounds exist one is binary (say A + B) and the other ternary (say A + 2B); if three, then one is binary and the others may be ternary (A ± 2B, and 2A + B), and so on.

00When binary compounds, or compounds of two elements, are decomposed by an electric current, the two elements make their appearance at opposite poles.

00If a compound contains two atoms it is termed a binary compound, if three a ternary, if four a quaternary, and so on.

00The nomenclature of acids follows the same general lines as that for binary compounds.

00Salts formed from hydracids terminate in -ide, following the rule for binary compounds.

00This view was accepted in 1817 by Leopold Gmelin, who, in his Handbuch der Chemie, regarded inorganic compounds as being of binary composition (the simplest being oxides, both acid and basic, which by combination form salts also of binary form), and organic compounds as ternary, i.e.

00The binary conception of compounds held by Berzelius received apparent support from the observations of Gay Lussac, in 1815, on the vapour densities of alcohol and ether, which pointed to the conclusion that these substances consisted of one molecule of water and one and two of ethylene respectively; and from Pierre Jean Robiquet and Jean Jacques Colin, showing, in 1816, that ethyl chloride (hydrochloric ether) could be regarded as a compound of ethylene and hydrochloric acid.

00a V(I - a) This constant k gives a numerical value for the chemical affinity, and the equation should represent the effect of dilution on the molecular conductivity of binary electrolytes.

00So the theory of the forms appertaining to a binary form of unrestricted order was first worked out by Cayley and P. A.

00A binary form which has a square factor has its discriminant equal to zero.

00THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables.

00Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are symbolical respreentations of the real coefficients Ã¯¿½o, ai,...

00To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a symbolic form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 Ã¯¿½ al = a;b4 -}-alb?

00By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1Ã¯¿½1 +a2m2) = aa a Ã¯¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alÃ¯¿½1+a2Ã¯¿½2) = a A Ã¯¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a Ã¯¿½ respectively.

00-When the binary form a y = (alxl +a2x2)n is transformed to A;.

00and we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a covariant is equally simple, because we see at once that since AE, B, Ce,...

00If f =ay, 4 = b' be any two binary forms, we generalize by forming the function (m-k)!

00- An important class of invariants, of several binary forms of the same order, was discovered by Sylvester.

00A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

00We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.

00Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.

00The Binary Quartic.-The fundamental system consists of five forms ax=f; (f,f')2=(ab) 2axbx=Ax; (f,f')4=(ab) 4= 2; (f, 0)1= (ao) azsi = (ab) 2 (cb) a:b x c5 =1; (f 4) 4 = (as) 4 = (6) 2 00 2 (ca) 2 = j, viz.

00-, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.

00For a further discussion of the binary sextic see Gordan, loc. cit., Clebsch, loc. cit.

00August von Gall in 1880 obtained the complete system of the binary octavic (Math.

0031-52, 1 391 5 2, 45 6); and, in 1888, that of the binary septimic, which proved to be much more complicated (Math.

00Single binary forms of higher and finite order have not been studied with complete success, but the system of the binary form of infinite order has been completely determined by Sylvester, Cayley, MacMahon and Stroh, each of whom contributed to the theory.

00As regards simultaneous binary forms, the system of two quadratics, and of any number of quadratics, is alluded to above and has long been known.

00The Binary Sextic.-The complete system consists of 26 forms, of which the simplest are x2y2z2 + (1 +8 m3) 2 (y3z3 +z3x3 +x3y3).

00These may be written, for the binary nie, Zka k _.

00The whole theory of invariants of a binary form depends upon the solutions of the equation SZ=o.

00Before discussing these it is best to trans form the binary form by substituting I !a i, 2 !

00One advantage we have obtained is that, if we now write ao =o, and substitute a 8 _ 1 for a,, when s>o, we obtain d d aO da l +al da 2 +a2 da Ã¯¿½....+an_2dan_1 which is the form of SZ for a binary (n- Henceby merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant of the same degree, and of weight w-8, appertaining to the (n-I) ic. Also, if we increase each suffix in a seminvariant, we obtain terms, free from a 0, of some seminvariant of degree 8 and weight w+8.

00The process is not applicable with complete success to quintic and higher ordered binary forms. This arises from the circumstance that the simple syzygies between the ground forms are not all independent, but are connected by second syzygies, and these again by third syzygies, and so on; this introduces new difficulties which have not been completely overcome.

00Perpetuants.-Many difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the (p1p2p3...) to where form of infinite order.

00A Similar Theorem Holds In The Case Of Any Number Of Binary Forms, The Mixed Seminvariants Being Derived From The Jacobians Of The Several Pairs Of Forms. If The Seminvariant Be Of Degree 0, 0' In The Coefficients, The Forms Of Orders P, Q Respectively, And The Weight W, The Degree Of The Covariant In The Variables Will Be P0 Qo' 2W =E, An Easy Generalization Of The Theorem Connected With A Single Form.

001 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.

00Restricted Substitutions We may regard the factors of a binary n ip equated to zero as denoting n straight lines through the origin, the co-ordinates being Cartesian and the axes inclined at any angle.

00Then a binary n", equated to zero, represents n straight lines through the origin, and the x, y of any line through the origin are given constant multiples of the sines of the angles which that line makes with two fixed lines, the axes of co-ordinates.

00Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

00By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, - 'el ' 'e2 ' 'e3 we may construct derived units of the third, fourth ...

00The perfecting of the microscope led to a full comprehension of the great doctrine of cell-structure and the establishment of the facts - (r) that all organisms are either, single corpuscles (so-called cells) of living material (microscopic animalcules, &c.) or are built up of an immense number of such units; (2) that all organisms begin their individual existence as a single unit or corpuscle of living substance, which multiplies by binary fission, the products growing in size and multiplying similarly by binary fission; and (3) that the life of a multicellular organism is the sum of the activities of the corpuscular units of which it consists, and that the processes of life must be studied in and their explanation obtained from an understanding of the chemical and physical changes which go on in each.

00In the Lavoisierian nomenclature acids were regarded as binary oxygenated compounds, the associated water being relegated to the position of a mere solvent.

00Points on the side AB will correspond to binary alloys containing only A and B, and so on.

00Interesting stars are: a Aurigae or Capella (the goat), one of the brightest stars in the heavens, determined by Newall and Campbell to be a spectroscopic binary; [3 Aurigae, a star of the second magnitude also a spectroscopic binary; e Aurigae, an irregularly variable star; and Nova Aurigae, a "new" star discovered by Anderson in 1892, and afterwards found on a photographic plate exposed at Harvard in December 1891.

00Ursae majoris is a beautiful binary star, its components having magnitudes 4 and 5; this star was one of the first to be recognized as a binary - i.e.

00But the fraction 1/128, or a continued binary division repeated seven times, is such a likely mode of rude subdivision that little stress can be laid on this.

00It has been connected theoretically with a binary division of the 10 shekels or "stone" of the Assyrian systems (28), 1290/16 being 80.625; this is suggested by the most usual multiples being 40 and 80 = 25 and 50 shekels of 129; it is thus akin to the mina of 50 shekels previously noticed.

00-- 100 feet, 66 feet or a chain of 100 links, rod, pole, or perch, measures from 10 feet to 1 foot; 18 inches; yard of 36 inches, 1/2, 1/4, 18, 1/16 yard, nail, inch, and duodecimal, decimal, and binary parts of an inch.

00These investigations have yielded many important discoveries, not only of new stars, and of large numbers of variable stars, but also of a wholly new class of double stars whose binary character is only revealed by peculiarities in their spectra.

00All Trypanosomes are capable of binary longitudinal fission, and this appears to be the chief method of multiplication.

00- Stages in Binary Longitudinal Fission of Trypanosoma brucei.

00These young individuals can themselves multiply by equal binary fission, giving Anterior flagellum; Posterior flagellum; Longitudinal striations nemes); Cytoplasmic vacuole.

00The parasites multiply in two ways - (a) by binary fission, and (b) by multiple division or segmentation.

00a, Ordinary forms of varying sizes; b c, stages in multiple division; d, binary fission; e, f, g, flagellate forms (after Rogers).

00Hence for a binary electrolyte the diffusion constant is measured by 2RT/F or 2UVRT/(U+V).

00Binary system >>

00The union of any two of the elements gave rise to binary compounds, such as oxides, acids, sulphides, &c. A substance containing three elements was a binary compound of the second order; thus salts, the most important compounds of this class, were formed by the union of acids and oxides, iron sulphate, for instance, being a compound of iron oxide with sulphuric acid.

00The most interesting members are: a Coronae, a binary consisting of a yellow star of the 6th magnitude, and a bluish star of the 7th magnitude; R Coronae, an irregular variable star; and T Coronae or Nova Coronae, a temporary or new star, first observed in 1866.

00There is reason to believe that all the stars of this class are binary systems, and that the variations of brightness are determined by the different aspects presented by the two component stars during the period of revolution.

00All the best-known short-period variables have been proved to be binary systems spectroscopically, and to have periods corresponding with the period of light variation, so that to this extent the hypothesis we have described is well founded; but it is doubtful if it is the whole explanation.

00In a number of cases measures of the relative positions of the two stars, continued for many years, have shown that they are revolving about a common centre; when this is so there can be no doubt that they form a binary system, and that the two components move in elliptic orbits about the common centre of mass, controlled by their mutual gravitation.

00They may thus be fairly regarded as constituting a binary system, though the gravitational attraction between some of the wider pairs must be very weak.

00The most rapid visual binary (leaving aside Capella for the moment) is Equulei, which completes a revolution in 5.7 years.

00Capella, whose period is only 104 days, was discovered to be double by means of the spectroscope, but has since been measured frequently as a visual binary at Greenwich.

00Far within the limit to which telescopic vision can extend binary systems are now being found by the spectroscope.

00Campbell one star in every seven examined is binary.

00We have necessarily to turn to binary systems for our data.

00Lewis, however, has shown that the fainter component of the binary system is often the more massive.

00He chose for his purpose the binary star 61 Cygni, which was the star with the most rapid apparent motion then known and therefore likely to be fairly near us, although only of the sixth magnitude.

00The method is of very limited application, for in general the orbital velocity of a visual binary is far too small to be found in this way; one of its first applications has been made to a Centauri, with the result that the parallax found in the ordinary way is completely confirmed.

00That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

00Scientific interest centres mainly on the following: - the nebula in Andromeda, one of the finest in the sky (see Nebula); y Andromedae, the finest binary in the heavens, made up of a yellow star of magnitude 2 2, and a blue-green of magnitude 51, the latter being itself binary; Nova Andromedae, a "new" star, discovered in the nebula by C. E.

00He announced in 1844 the binary character of Sirius and Procyon from their disturbed proper motions; and was preparing to attack the problem solved later by the discovery of Neptune, when fatal illness intervened.

00Where there is a division into sixteen parts, a binary scale may be formed by dividing into groups of two, four or eight.

00up to 2 lb give the basis for a binary scale up to not more than eight figures, only o and 1 being used.

00The points of the compass might similarly be expressed by numbers in a binary scale; but the numbers would be ordinal, and the expressions would be analogous to those of decimals rather than to those of whole numbers.

00Iob where a is less than io, and the index b is expressed in the binary scale, e.g.

00The British method is a mixture of the last two, but with an index-scale which is partly ternary and partly binary.

00OXIDE, in chemistry, a binary compound of oxygen and other elements.

00Binary fission has been widely observed, both in the active condition or after loss of the flagella: it differs from that of true Flagellates in not being longitudinal, but transverse or oblique (fig.

00We will have more to say about this binary pulsar later in the course.

00The binary X-ray pulsar Cen X-3 was observed over an orbital eclipse, and time variation of the iron line structure was studied.

00I compare every binary quantic with a chemical atom...

00Vanity and self-pity form a binary, and so can switch between each other.

00Suppose that the self-pity mode changes to its binary, vanity.

00For synchronization, simple locks (a.k.a. mutexes or binary semaphores) are provided.

00LO - Load binary file Format: LO filename The load binary file command loads the file specified by filename into memory.

00This structuralist analysis of texts tends to focus on binary oppositions.

00This only shows the need to use more subdivisions instead binary oppositions.

00Binary: A plant in which hot water flows through heat exchangers and causes a fluid to boil.

00Binary cycle - Hot water gets pumped through a heat exchanger which heats up a liquid with a low boiling point such as butane.

00As is the case with many handbags, perhaps Bag Borrow or Steal is best dealt with in the tried and true binary of women's fashion and accessories: you either love it, or hate it.

00But the fraction 1/128, or a continued binary division repeated seven times, is such a likely mode of rude subdivision that little stress can be laid on this.

01It has been connected theoretically with a binary division of the 10 shekels or "stone" of the Assyrian systems (28), 1290/16 being 80.625; this is suggested by the most usual multiples being 40 and 80 = 25 and 50 shekels of 129; it is thus akin to the mina of 50 shekels previously noticed.

01These investigations have yielded many important discoveries, not only of new stars, and of large numbers of variable stars, but also of a wholly new class of double stars whose binary character is only revealed by peculiarities in their spectra.

01All Trypanosomes are capable of binary longitudinal fission, and this appears to be the chief method of multiplication.

01- Stages in Binary Longitudinal Fission of Trypanosoma brucei.

01These young individuals can themselves multiply by equal binary fission, giving Anterior flagellum; Posterior flagellum; Longitudinal striations nemes); Cytoplasmic vacuole.

01The parasites multiply in two ways - (a) by binary fission, and (b) by multiple division or segmentation.

01The most interesting members are: a Coronae, a binary consisting of a yellow star of the 6th magnitude, and a bluish star of the 7th magnitude; R Coronae, an irregular variable star; and T Coronae or Nova Coronae, a temporary or new star, first observed in 1866.

01There is reason to believe that all the stars of this class are binary systems, and that the variations of brightness are determined by the different aspects presented by the two component stars during the period of revolution.

01All the best-known short-period variables have been proved to be binary systems spectroscopically, and to have periods corresponding with the period of light variation, so that to this extent the hypothesis we have described is well founded; but it is doubtful if it is the whole explanation.

01In a number of cases measures of the relative positions of the two stars, continued for many years, have shown that they are revolving about a common centre; when this is so there can be no doubt that they form a binary system, and that the two components move in elliptic orbits about the common centre of mass, controlled by their mutual gravitation.

01They may thus be fairly regarded as constituting a binary system, though the gravitational attraction between some of the wider pairs must be very weak.

01The most rapid visual binary (leaving aside Capella for the moment) is Equulei, which completes a revolution in 5.7 years.

01Capella, whose period is only 104 days, was discovered to be double by means of the spectroscope, but has since been measured frequently as a visual binary at Greenwich.

01Far within the limit to which telescopic vision can extend binary systems are now being found by the spectroscope.

01Campbell one star in every seven examined is binary.

01We have necessarily to turn to binary systems for our data.

01Lewis, however, has shown that the fainter component of the binary system is often the more massive.

01He chose for his purpose the binary star 61 Cygni, which was the star with the most rapid apparent motion then known and therefore likely to be fairly near us, although only of the sixth magnitude.

01The method is of very limited application, for in general the orbital velocity of a visual binary is far too small to be found in this way; one of its first applications has been made to a Centauri, with the result that the parallax found in the ordinary way is completely confirmed.

01That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

01He announced in 1844 the binary character of Sirius and Procyon from their disturbed proper motions; and was preparing to attack the problem solved later by the discovery of Neptune, when fatal illness intervened.

01up to 2 lb give the basis for a binary scale up to not more than eight figures, only o and 1 being used.

01The points of the compass might similarly be expressed by numbers in a binary scale; but the numbers would be ordinal, and the expressions would be analogous to those of decimals rather than to those of whole numbers.

01Iob where a is less than io, and the index b is expressed in the binary scale, e.g.

01In addition to this there are some more unusual display options, including binary codes.

01C3PO has knocked about the galaxy a bit, and his memory may or may not have been altered once or twice, but he's picked up some odd information along the way, such as programming binary load lifters, whatever they are.

01Setting this parameter to TRUE indicates that the string will be converted into a raw sixteen character binary format.

01Setting this parameter to TRUE indicates that the string will be converted into a raw twenty character binary format.

01

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