# How to use *Theorem* in a sentence

This is the true standpoint from which the

**theorem**should be regarded.This enabled David Hilbert to produce a very simple unsymbolic proof of the same

**theorem**.By means of this

**theorem**it can be shown that, whatever the value of n may be, f 1 + (plq)(i)x+(p/q)(2)x2+...In general we can prove in the same way the -

**Theorem**.Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor theorems of great elegance, - among which may be mentioned his

**theorem**that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum.AdvertisementThe binomial

**theorem**gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.The binomial

**theorem**for positive integral index may then be written (x + y) n = -iyi +.The argument involves the

**theorem**that, if 0 is a positive quantity less than I, 0 t can be made as small as we please by taking t large enough; this follows from the fact that tlog 0 can be made as large (numerically) as we please.This is commonly called Stokes's

**theorem**.Legendre, in 1783, extended Maclaurin's

**theorem**concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the**theorem**was equally true for any confocal ellipsoids.AdvertisementThe first formal proof of Lagrange's

**theorem**for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality.If we represent this expression by f (x), the expression obtained by changing x into x-+-h is f(x+h); and each term of this may be expanded by the binomial

**theorem**.The relation, when written in the form (23), is known as Vandermonde's

**theorem**.It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homogeneous light, and that, in accordance with Fourier's

**theorem**, each homogeneous component of a mixture may be treated separately.Thus, if a= d, we should have 1=2+4+77(49425+...) which is true by a known

**theorem**.AdvertisementA full discussion would call for the formal application of Fourier's

**theorem**, but some conclusions of importance are almost obvious.This

**theorem**was published in 1643, at the end of his treatise De motu gravium projectorum, and it was confirmed by the experiments of Raffaello Magiotti on the quantities of water discharged from different ajutages under different pressures (1648).The

**theorem**of Torricelli was employed by many succeeding writers, but particularly by Edme Mariotte (1620-1684), whose Traite du mouvement des eaux, published after his death in the year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at Versailles and Chantilly.This

**theorem**is called generally the principle of Archimedes.As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a

**theorem**of Archimedes, has the same volume as the cone ORR', or rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside FIG.AdvertisementIn particular, for a jet issuing into the atmosphere, where p=P, q 2 /2g = h - z, (9) or the velocity of the jet is due to the head k-z of the still free surface above the orifice; this is Torricelli's

**theorem**(1643), the foundation of the science of hydrodynamics.The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical

**theorem**that aw is constant for all time, and the same for every cross-section of the vortex filament.The binomial

**theorem**is a celebrated**theorem**, originally due to Sir Isaac Newton, by which any power of a binomial can be expressed as a series.In its modern form the

**theorem**, which is true for all values of n, is written as (x +a) n -1+ I.It is a fundamental

**theorem**in attractions that a thin spherical shell of matter which attracts according to the potential law of the inverse square acts on all external points as of a if it were concentrated at its centre.AdvertisementStokes's

**theorem**becomes an obvious truism if applied to an incompressible fluid.Let us apply the above

**theorem**to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).Hence the total flux is - (+ d2V d 2 V d2V dye + dz2) dy dz, dx2 and by the previous

**theorem**this must be equal to 4'rrp dxdydz.Hence if dS and dS' are the areas of the ends, and +E and - E' the oppositely directed electric forces at the ends of the tube, the surface integral of normal force on the flux over the tube is EdS - E'dS' (20), and this by the

**theorem**already given is equal to zero, since the tube includes no electricity.We begin with a general dynamical

**theorem**, whose special application, when the dynamical system is identified with a gas, will appear later.AdvertisementBut Landen's capital discovery is that of the

**theorem**known by his name (obtained in its complete form in the memoir of 1775, and reproduced in the first volume of the Mathematical Memoirs) for the expression of the arc of an hyperbola in terms of two elliptic arcs.The formula applied can then be either Simpson's rule or a rule based on Gauss's

**theorem**for two ordinates (§ 56).Fourier's

**theorem**asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...Fourier's

**theorem**can also be usefully applied to the disturbance of a source of sound under certain conditions.Now we may resolve these trains by Fourier's

**theorem**into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.AdvertisementIf then we resolve Ahbkc into harmonics by Fourier's

**theorem**, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant.We see, then, that the conditions for the application of Fourier's

**theorem**are equivalent to saying that all disturbances will travel along the system with the same velocity.In many vibrating systems this does not hold, and then Fourier's

**theorem**is no longer an appropriate resolution.But with the sole exception of proving that the volumes of spheres are in the triplicate ratio of their diameters, a

**theorem**probably due to Eudoxus, no mention is made of its mensuration.To Legendre is due the

**theorem**known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."AdvertisementLegendre shows that Maclaurin's

**theorem**with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution.The third memoir relates to Laplace's

**theorem**respecting confocal ellipsoids.The best known of these, which is called Legendre's

**theorem**, is usually given in treatises on spherical trigonometry; by means of it a small spherical triangle may be treated as a plane triangle, certain corrections being applied to the angles.Legendre's

**theorem**is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.A remarkable

**theorem**is I -x.AdvertisementThe

**theorem**for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form I - cos 0 = 2 sin e 20.With the mention of the Porisms of Euclid we have an account of the relation of porism to

**theorem**and problem.To the former belong the theorems (t), (2), and (3), and to the latter especially the

**theorem**(4), and also, probably, his solution of the two practical problems. We infer, then, [t] that Thales must have known the**theorem**that the sum of the three angles of a triangle are equal to two right angles.No doubt we are informed by Proclus, on the authority of Eudemus, that the

**theorem**Euclid i.The

**theorem**, then, seems to have been arrived at by induction, and may have been suggested by the contemplation of floors or walls covered with tiles of the form of equilateral triangles, or squares, or hexagons.AdvertisementThe knowledge of this

**theorem**is distinctly attributed to Thales by Plutarch, and it was probably made use of also in his determination of the distance of a ship at sea.With Locke, Hume professes to regard this problem as virtually covered or answered by the fundamental psychological

**theorem**; but the superior clearness of his reply enables us to mark with perfect precision the nature of the difficulty inherent in the attempt to regard the two as identical.In this way he established the famous

**theorem**that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear.Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's

**theorem**(1730).By imagining the successive positions to be taken infinitely close to one another we derive the

**theorem**stated.A geometrical proof of this

**theorem**, which is not restricted to a two-dimensional system, is given later (If).The

**theorem**that any coplanar system of forces can be reduced to a force acting through any assigned point, together with a couple, has an important illustration in the theory of the distribution of shearing stress and bending moment in a horizontal beam, or other structure, subject to vertical extraneous forces.This is essentially a

**theorem**of projective geometry, but the following statical proof is interesting.This

**theorem**enables us, when one funicular has been drawn, to construct any other without further reference to the force-diagram.This

**theorem**is due to 0.It follows from Eulers

**theorem**that the most general displacement of a rigid body may be effected by a pure translation which brings any one point of it to its final position 0, followed by a pure rotation about some axis through 0.The preceding

**theorem**, which is due to Michel Chasles (1830), may be proved in various other interesting ways.The analogy between the mathematical relations of infinitely small displacements on the one hand an-d those of force-systems on the other enables us immediately to convert any

**theorem**in the one subject into a**theorem**in the other.This,

**theorem**, also due to Lagrange, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances.There is also an analogue to Lagranges Second

**Theorem**, viz.In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the

**theorem**of 7 will then apply in every respect.By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that

**theorem**to each of them, with the aid, if necessary, of Moseleys principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.The above construction for Z is a corollary of the general

**theorem**given in 127.An important

**theorem**contained in it is known as Green's**theorem**, and is of great value.The following

**theorem**covers a large number of important cases.Whatever the deformation of the originally straight boundary of the axial section may be, it can be resolved by Fourier's

**theorem**into deformations of the harmonic type.One of the earliest was devoted to electrical conduction in a thin plate, and especially in a circular one, and it also contained a

**theorem**which enables the distribution of currents in a network of conductors to be ascertained.Nevertheless it is only human nature, to derive some pleasure from being cited, now and then, even about a `

**Theorem**'; especially where.This is true, whatever the arrangement of the original objects may be, and wherever the new one is introduced; and therefore, if the

**theorem**is true for 8, it is true for 9.More generally, if we have obtained a as an approximate value for the pth root of N, the binomial

**theorem**gives as an approximate formula p,IN =a+6, where N = a P + pap - 19.In 1829 he discovered the

**theorem**, regarding the determination of the number of real roots of a numerical equation included between given limits, which bears his name (see Equation, V.), and in the following year he was appointed professor of mathematics at the College Rollin.It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the

**theorem**in its universality, it is necessary to take account of these various circumstances.Trilinear and Tangential Co-ordinates.---The Geometrie descriptive, by Gaspard Monge, was written in the year 1794 or 1 795 (7th edition, Paris, 1847), and in it we have stated, in piano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental

**theorem**of reciprocal polars, viz.The

**theorem**is here referred to partly on account of its bearing on the theory of imaginaries in geometry.Stating the

**theorem**in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the**theorem**is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other.The well-known

**theorem**as to radical axes may be stated as follows.It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the

**theorem**that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m - 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.It is to be understood that the

**theorem**extends to all points or lines, drawn or not drawn; thus if in the first figure there are any number of points on a line drawn or not drawn, the corresponding lines in the second figure, produced if necessary, must meet in a point.And we thus see how the

**theorem**extends to curves, their points and tangents; if there is in the first figure a curve of the order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that is, the corresponding curve must be of the class in.The

**theorem**of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in the Memoire de geometrie sur deux principes generaux de la science, la dualite et l'homographie, which forms a sequel to the Apercu historique l'origine t.It is implied in Pliicker's

**theorem**that, m, n, signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, will have the same significations, viz.The whole theory of the inflections of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x +y +z +6lxyz= o; and in particular a proof is given of Plucker's

**theorem**that the nine points of inflection of a cubic curve lie by threes in twelve lines.For real figures we have the general

**theorem**that imaginary intersections, &c., present themselves in conjugate pairs; hence, in particular, that a curve of an even order is met by a line in an even number (which may be = o) of points; a curve of an odd order in an odd number of points, hence in one point at least; it will be seen further on that the**theorem**may be generalized in a remarkable manner.Stated in regard to the cone, we have there the fundamental

**theorem**that there are two different kinds of sheets; viz., the single sheet, not separated into two parts by the vertex (an instance is afforded by the plane considered as a cone of the first order generated by the motion of a line about a point), and the double or twin-pair sheet, separated into two parts by the vertex (as in the cone of the second order).And we have then the

**theorem**, two odd circuits intersect in an odd number of points; an odd and an even circuit, or two even circuits, in an even number of points.The general

**theorem**is that two curves corresponding rationally to each other have the same deficiency.In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the

**theorem**is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.And so if D =2, then the transformed curve is a nodal quartic; 4 can be expressed as the square root of a sextic function of 0 and the

**theorem**is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a sextic function of 0.It is a form of the

**theorem**for the case D = r, that the coordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be r -0 2 .r - k 2 0 2, and writing 8 =snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.The

**theorem**of united points in regard to points in a right line was given in a paper, June-July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), " Sur les courbes planes ou a double courbure dont les points peuvent se determiner individuellement - application du principe de correspondance dans la theorie de ces courbes."The principle of correspondence, or say rather the

**theorem**of united points, is a most powerful instrument of investigation, which may be used in place of analysis for the determination of the number of solutions of almost every geometrical problem.And he gives the

**theorem**, a system of conics satisfying four conditions, and having the characteristics (µ, v) contains 2v - µ line-pairs (that is, conics, each of them a pair of lines), and point-pairs (that is, conics, each of them a pair of points, - coniques infiniment aplaties), which is a fundamental one in the theory.The rising revolutionary school in France, if they had read it, would have taken it for a demonstration of the

**theorem**to be proved.To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding

**theorem**for the multiplication of two matrices.Laplace developed a

**theorem**of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.To Gauss is due the establishment of the important

**theorem**, that the product of two determinants both of the second and third orders is a determinant.What is noteworthy in this

**theorem**is that this relation depends only on the sum of the masses.His

**theorem**that a fluid issues from a small orifice with the same velocity (friction and atmospheric resistance being neglected) which it would have acquired in falling through the depth from its surface is of fundamental importance in hydraulics.If five points be given, Pascal's

**theorem**affords a solution; if five tangents, Brianchon's**theorem**is employed.A connexion between the number of faces, vertices and edges of regular polyhedra was discovered by Euler, and the result, which assumes the form E + 2' = F ± V, where E, F, V are the number of edges, faces and vertices, is known as Euler's

**theorem**on polyhedra.We find Newton's

**theorem**, that "action and reaction are equal and opposite," stated with approximate precision in his treatise Della scienza meccanica, which contains the substance of lectures delivered during his professorship at Padua; and the same principle is involved in the axiom enunciated in the third of his mechanical dialogues, that "the propensity of a body to fall is equal to the least resistance which suffices to support it."Desargues has a special claim to fame on account of his beautiful

**theorem**on the involution of a quadrangle inscribed in a conic. Pascal discovered a striking property of a hexagon inscribed in a conic (the hexagrammum mysticum); from this**theorem**Pascal is said to have deduced over 400 corollaries, including most of the results obtained by earlier geometers.He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's

**theorem**rigorously, establishing his well-known form of the remainder.Fermat's

**Theorem**, if p is prime and a is prime to p then a p-1 -1 is divisible by p, was first given in a letter of 1640.If we make large enough to expand the numerator using the binomial

**theorem**(so that behaves as ), then as.A similar proof using discrete Abel summation delivers a variant of the

**theorem**in terms of ls instead of li.Their

**theorem**prover is written in the functional programming language lisp which is also the language in which theorems are represented.The prime number

**theorem**shows that if we pick an integer n at random, it will be prime with probability.The

**theorem**was first propounded by F Guthrie in 1853.Apply Nyquist's stability

**theorem**, to predict closed-loop stability from open-loop Nyquist or Bode diagrams.We develop a theory of semantic tableaux for BI, thereby providing an elegant basis for efficient

**theorem**proving tools for BI.I proved, for example, a generalized coverage

**theorem**that specialized to both the known complete lattice and preframe versions.Example 1.. 15 Use the binomial

**theorem**to expand (x + y) 5.My favorite school memory My favorite memory from school is the Pythagorean

**theorem**.A famous 1996 no-go

**theorem**by Lo, Chau, and Mayers claims that no such protocol exists, but has been repeatedly challenged.However, the significance of a mathematical

**theorem**is relative to its contexts.The goal is to prove the prime number

**theorem**in the form.There is a

**theorem**proved by Kurt Godel in 1931, which is the Incompleteness**Theorem**for mathematics.Kreisel's suggestion was taken up with great success by Barwise [1967 ], where his compactness

**theorem**was proved.In MV they take some of these subjects further and include chapters on the four-colour

**theorem**, Ramsey theory, Catalan numbers and more.Karp's partial isomorphism

**theorem**was first proved in Karp [1965]; see also Barwise [1973] .This completeness

**theorem**turns out to be equivalent to the axiom scheme of replacement.Continue geometry, including Pythagoras '

**Theorem**and basic trigonometry.This

**theorem**has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.Between 1886 and 1892 he published a series of papers on the foundations of the kinetic theory of gases, the fourth of which contained what was, according to Lord Kelvin, the first proof ever given of the Waterstdn-Maxwell

**theorem**of the average equal partition of energy in a mixture of two different gases; and about the same time he carried out investigations into impact and its duration.On the other hand, Thorold Rogers, not to speak of earlier objectors, described the law as a " dismal and absurd

**theorem**."This is a particular case of Taylor's

**theorem**(see Infinitesimal Calculus).Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see Circle), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the binomial

**theorem**, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions.The multi- (or poly-) nomial

**theorem**has for its object the expansion of any power of a multinomial and was discussed in 1697 by Abraham Demoivre (see Combinatorial Analysis).This work included the "Logometria," the trigonometrical

**theorem**known as "Cotes'**Theorem**on the Circle" (see TRIGONOMETRY), his**theorem**on harmonic means, subsequently developed by Colin Maclaurin, and a discussion of the curves known as "Cotes' Spirals," which occur as the path of a particle described under the influence of a central force varying inversely as the cube of the distance.Any periodic curve may be resolved into sine or harmonic curves by Fourier's

**theorem**.If we form all the partitions of 6 into not more than three parts, these are 6, 51, 4 2, 33, 411, 321, 222, and the conjugates are Iiiiii, 21iii, 221i, 222, 311i, 321, 33, where no part is greater than 3; and so in general we have the

**theorem**, the number of partitions of n into not more than k parts is equal to the number of partitions of n with no part greater than k.He attempted the quadrature of the circle by interpolation, and arrived at the remarkable expression known as Wallis's

**Theorem**(see Circle, Squaring Of).The orthocentre of a triangle circumscribing a parabola is on the directrix; a deduction from this

**theorem**is that the centre of the circumcircle of a self-conjugate triangle is on the directrix ("Steiner's**Theorem**").By means of this

**theorem**we can show that the previous reduction of any system to a wrench is unique.The idea of such forces, however, had been distinctly formed by Newton, who gave the first example of the calculation of the effect of such forces in his

**theorem**on the alteration of the path of a light-corpuscle when it enters or leaves a dense body.Samuel Hopkins laid even greater stress than Edwards on the

**theorem**that virtue consists in disinterested benevolence; but he went counter to Edwards in holding that unconditional resignation to God's decrees, or more concretely, willingness to be damned for the glory of God, was the test of true regeneration; for Edwards, though often quoted as holding this doctrine, protested against it in the strongest terms. Hopkins, moreover, denied Edwards's identity theory of original sin, saying that our sin was a result of Adam's and not identical with it; and he went much further than Edwards in his objection to " means of grace," claiming that the unregenerate were more and more guilty for continual rejection of the gospel if they were outwardly righteous and availed themselves of the means of grace.The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves; Newton gives (in two short paragraphs without any development) the remarkable

**theorem**that the five divergent parabolas by their shadows generate and exhibit all the cubic curves.This bold attempt is entirely factitious and verbal, and it is only his employment of various terms not generally used in such a connexion (axiom,

**theorem**, corollary, etc.) that gives his treatise' its apparent originality.All these measures work by setting up a set of numerical recursion equations derived from the input clauses to the

**theorem**prover.We can see how we might almost have expected GÃ¶del 's

**theorem**to distinguish self-conscious beings from inanimate objects.Apply Nyquist 's stability

**theorem**, to predict closed-loop stability from open-loop Nyquist or Bode diagrams.Since is itself a summation function, our main objective, the prime number

**theorem**, is a result of exactly this type.Of course, the two are linked through the fundamental

**theorem**of calculus.Gauss 's dissertation was a discussion of the fundamental

**theorem**of algebra.Using a

**theorem**prover provides a natural way of dealing with this.These tools have ranged from fault tree packages through to what some might consider exotic

**theorem**provers.Using tactics is not a novel idea, particularly in the area of

**theorem**proving.Kreisel 's suggestion was taken up with great success by Barwise [1967 ], where his compactness

**theorem**was proved.Even after seeing some proof, Alyssa continued doubting the

**theorem**.And, true to the filtration

**theorem**, smaller designers have taken lengths to knockoff the "IT".With Descartes the use of exponents as now employed for denoting the powers of a quantity becomes systematic; and without some such step by which the homogeneity of successive powers is at once recognized, the binomial

**theorem**could scarcely have been detected.Another curious

**theorem**proposed by Bouilland in 1625 as a substitute for Kepler's second law is that the angular motion of the body as measured around the empty focus F' is (approximately) uniform.The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important

**theorem**of expressibility is, directly we exclude unity from the partitions, a**theorem**concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.Hence the

**theorem**of expressibility enunciated above.The similar

**theorem**for n systems of quantities can be at once written down.There are extensions of the binomial

**theorem**, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.Thus the ternary quartic is not, in general, expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a

**theorem**due to Sylvester.We may, by a well-known

**theorem**, write the result as a coefficient of z w in the expansion of 1 - z n+1.Hence the

**theorem**is established.This is the binomial

**theorem**for a positive integral index.This

**theorem**is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed.We therefore have the fundamental

**theorem**that the angular velocity of the body around the centre of attraction varies inversely as the square of its distance, and is therefore at every point proportional to the gravitation of the sun.