# Theorem sentence example

theorem
• 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.
• In general we can prove in the same way the - Theorem.
• It has been mentioned in ï¿½ 41 (ix.) that the binomial theorem can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account.
• 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+...
• = exp,udl where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem.
• Now the symbolic expression of the seminvariant can be expanded by the binomial theorem so as to be exhibited as a sum of products of seminvariants, of lower degrees if alai 0-2a2 +...+crea0 can be broken up into any two portions (alai -1-0-2a2-1-ï¿½ï¿½ï¿½ +asas) +(as+1as +1 +o-8+2as+2+ï¿½ï¿½ï¿½ +ooae), such that Q1 +a2+...
• 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.
• The 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.
• Then, provided a r includes the greatest term, it will be found that (A - a)" lies between 0' r and ar+1ï¿½ For actual calculation it is most convenient to write the theorem in the form methods of procedure.
• The binomial theorem for positive integral index may then be written (x + y) n = -iyi +.
• These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in ï¿½ 52 (vi.).
• 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.
• (viii.) In applying the theorem to concrete cases, conversion of a number into a continued fraction is often useful.
• (2) A theorem relating to the apparent curvature of the geocentric path of a comet.
• 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.
• The 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.
• (iv.) The procedure is sometimes stated differently, the transposition being regarded as a corollary from a general theorem that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation.
• (ii.) We can prove the theorem of ï¿½ 41 (v.) by a double application of the method.
• (c) Thus, if the theorem of ï¿½ 41 (v.) is true for r= p, it is true for r= p+1.
• 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.
• 2 m - 1 m 2sin ' a formula which may be verified by Fourier's theorem.
• A 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.
• In 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.
• - I.n - 2 asxn.-3 The I.2 I.2.3 reader is referred to the article Algebra for the proof and applications of this theorem; here we shall only treat of the history of its discovery.
• &c., where p+pq is the quantity whoseTi power or root is required, p the first term of that quantity, and q the quotient of the rest divided by p, m the power, which may be a positive or negative integer or a fraction, and a, b, c, &c., the several terms in order, In a second letter, dated the 24th of October 1676, to Oldenburg, Newton gave the train of reasoning by which he devised the theorem.
• 2 3 4 The binomial theorem was thus discovered as a development of John Wallis's investigations in the method of interpolation.
• 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.
• We have then a very important theorem as follows: - If any closed surface be described in an electric field which wholly encloses or wholly excludes electrified bodies, then the total flux through this surface is equal to 47r - times the total quantity of electricity within it.'
• This is commonly called Stokes's theorem.
• Stokes'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.
• § 99a (3rd ed., 1892), where the expression in question is deduced as a corollary of Green's theorem.
• We begin with a general dynamical theorem, whose special application, when the dynamical system is identified with a gas, will appear later.
• Considering only those states of the system which have a given value of E2, it can be proved, as a theorem in pure mathematics,' that when s, s', ...
• But 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.
• If 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."
• Legendre 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.
• The 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.
•  We see also in the theorem (4) the first trace of the important conception of geometrical loci, which we, therefore, attribute to Thales.
•  Thales discovered the theorem that the sides of equiangular triangles are proportional.
• The 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.
• In a scientific point of view: (a) we see, in the first place, that by his two theorems he founded the geometry of lines, which has ever since remained the principal part of geometry; (b) he may, in the second place, be fairly considered to have laid the foundation of algebra, for his first theorem establishes an equation in the true sense of the word, while the second institutes a proportion.'
• 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.
• For Demoivre's Theorem see Trigonometry: Analytical.
• 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.
• The proof is similar to that of the corresponding theorem of plane kinematics (~ 3).
• 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.
• (6) If we put ~, ~Y, s=o, the theorem is proved as regards axes parallel to Ox.
• We find, exactly as in the proof of Lagranges First Theorem (~ If),, that 3/4~(m.OV) =1/8~(m) OK2+1/8~(m.KV2); (8)
• There is also an analogue to Lagranges Second Theorem, viz.
• The typical case is where the extraneous forces are of the simple-harmonic type cos (at+~); the most general law of variation with time can be derived from this by superposition, in virtue of Fouriers theorem.
• Then from the proportionality and parallelism sides of a triangle, there results the following of the load and the two resistances applied to each piece of the structure to the three theorem (originally due to Rankine): If from the angles of the polygon of loads there be drawn lines (Ri, R2, &c.), each of which is parallel to the resistance (as Pi F2, &c.) exerted FIG.
• 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.
• Principle of the Transformation of Structuret.~-Here we have the following theorem: If a structure of a given figure have stability of position .inder a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first system of lines.
• 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 elementary theory of optical systems leads to the theorem: Rays of light proceeding from any " object point " unite in an " image point "; and therefore an " object space " is reproduced in an " image space."
• 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.
• The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.
• 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.
• The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in either figure there correspond in the other figure a line and point respectively.
• 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.
• A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is =4; it follows that the quartic has at most 8 real inflections.
• 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.
• - The theorem is obtained very easily from the last preceding definition of a determinant.
• 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.
• (3) Whewell's theorem: if a point R be taken at a distance from the sun equal to the major axis of the orbit of a planet and, therefore, at double the mean distance of the planet, the speed of the latter at any point is equal to the speed which a body would acquire by falling from the point R to the actual position of the planet.
• 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.
• Theorem 1: Unique Correctness Each hexagram has a unique correctness hexagram.
• 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 ; see also Barwise  .
• 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."
• From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, Ã¯¿½Ã¯¿½Ã¯¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.
• Now by the expansion theorem the determinant becomes (-)1 +2+3+Ã¯¿½.Ã¯¿½+2nB.0 = (- I)n(2n +1) +nC =C.
• If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +Ã¯¿½Ã¯¿½Ã¯¿½ +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.
• In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " Ã¯¿½Ã¯¿½ ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively.
• Now the symbolic expression of the seminvariant can be expanded by the binomial theorem so as to be exhibited as a sum of products of seminvariants, of lower degrees if alai 0-2a2 +...+crea0 can be broken up into any two portions (alai -1-0-2a2-1-Ã¯¿½Ã¯¿½Ã¯¿½ +asas) +(as+1as +1 +o-8+2as+2+Ã¯¿½Ã¯¿½Ã¯¿½ +ooae), such that Q1 +a2+...
• (ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of pa= qb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions.
• Then, provided a r includes the greatest term, it will be found that (A - a)" lies between 0' r and ar+1Ã¯¿½ For actual calculation it is most convenient to write the theorem in the form methods of procedure.
• (ii.) We can prove the theorem of Ã¯¿½ 41 (v.) by a double application of the method.
• (c) Thus, if the theorem of Ã¯¿½ 41 (v.) is true for r= p, it is true for r= p+1.
• This is a particular case of Taylor's theorem (see Infinitesimal Calculus).
• These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in Ã¯¿½ 52 (vi.).
• It has been mentioned in Ã¯¿½ 41 (ix.) that the binomial theorem can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account.
• Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (Ã¯¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.
• 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.
• (This solution may be verified in the same manner as Poisson's theorem, in which k=o.) We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of co-ordinates that the rotations are to be estimated.
• 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).
• Next he applied his theorem 4 BO+OA: AB:: OB: BD to calculate BD; from this in turn he calculated the semi-sides of the circumscribed regular 24-gon, 48-gon and 96-gon, and so finally established for the circumscribed regular 96-gon that perimeter: diameter 3+V :I.
• The proof of the correctness of the construction is seen to be involved in the following theorem, which serves likewise to throw new light on the subject: - AB being any straight line whatever, and the above construction being made, then AB is the diameter of the circle circumscribed by the square AB CD (self-evident), AB, is the diameter of the circle circumscribed by the regular 8-gon having the same perimeter as the square, AB, is the diameter of the circle circumscribed by the regular 16-gon having the same perimeter as the square, and so on.
• 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.
• This paper is principally based on the following general theorem, which is a remarkable extension of Pascal's hexagram: "If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, &c., respectively, then the free summit moves on a curve of the degree 2mnp. ..
• 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.
• +amam Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of al, a2, ...a m, which arise, in terms of b1, b2, ...'
• From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ï¿½ï¿½ï¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.
• Now by the expansion theorem the determinant becomes (-)1 +2+3+ï¿½.ï¿½+2nB.0 = (- I)n(2n +1) +nC =C.
• If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +ï¿½ï¿½ï¿½ +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.
• Hence the theorem of expressibility enunciated above.
• 1 1 where laan and di denotes, not s successive operations of d1, but the operator of order s obtained by raising d l to the s th power symbolically as in Taylor's theorem in the Differential Calculus.
• The similar theorem for n systems of quantities can be at once written down.
• To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.
• In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " ï¿½ï¿½ ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively.
• 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.
• Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (ï¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.
• 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.
• If we expand the symbolic expression by the multinomial theorem, and remember that any symbolic product ai 1 a2 2 a3 3 ...
• 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.