# Differential Sentence Examples

Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with

**differential**forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.Suppose the key to be depressed, then a current flows through one winding of the

**differential**relay to line and through the other winding and rheostat to earth.This arrangement can be duplexed in the way already explained, by providing

**differential**relays and arranging for the outgoing currents to divide differentially through the two relays at each end.The series of equations of the type (3) is then replaced by a single linear partial

**differential**equation, or by a set of two or three such equations, according to the number of dependent variables.Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial

**differential**coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial**differential**coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy dz dx / leading to the

**differential**equation of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy + dz with two similar equations.Expressed as a

**differential**relation, with the value of c-TKL ai,G+2 (a 2 +A) a d ?The partial

**differential**coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial

**differential**coefficient with respect to a momentum component will give the component of velocity to correspond.This is mathematically expressed by the statement that dE is an exact

**differential**of a function of the co-ordinates defining the state of the body, which can be integrated between limits without reference to the relation representing the path along which the variations are taken.AdvertisementThis inequality holds in all cases, but cannot in general be applied to an irreversible change, because Od4 is not a perfect

**differential**, and cannot be evaluated without a knowledge of the path or process of transformation.The value of Od4 is then the same as d(64), which is a perfect

**differential**, so that the condition may be written d(46-E) =dW.In this case dW=pdv=d(pv), a perfect

**differential**, so that the external work done is known from the initial and final states.Meanwhile the study of mathematics was not neglected, as appears not only from his giving instruction in geometry to his younger brother Daniel, but from his writings on the

**differential**, integral, and exponential calculus, and from his father considering him, at the age of twenty-one, worthy of receiving the torch of science from his own hands.The hypothesis that the state was steady, so that interchanges arising from convection and collisions of the molecules produced no aggregate result, enabled him to interpret the new constants involved in this law of distribution, in terms of the temperature and its spacial

**differential**coefficients, and thence to express the components of the kinetic stress at each point in the medium in terms of these quantities.AdvertisementIf we divide throughout by T, we obtain d 2 (n+3)RN T +RNd v, showing that dQ/T is a perfect

**differential**.In what follows it will be assumed that the conditions of continuity (which imply the continuity not only of u but also of some of its

**differential**coefficients) are satisfied, subject to the small errors in the values of u actually given; the limits of these errors being known.The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the

**differential**coefficients u', u", u"',.The following are the results (for the formulae involving chordal areas), given in terms of

**differential**coefficients and of central differences.At the same time the

**differential**duty on refined sugar, which operated as protection to the sugar trust, was not abolished, as the ardent tariff reformers had proposed, but kept in substance not greatly changed.AdvertisementHe published a memoir on the integration of partial

**differential**equations and a few others, which have not been noticed above, but they relate to subjects with which his name is not especially associated.Pfaff's researches bore chiefly on the theory of series, to which he applied the methods of the so-called combinatorial school of German mathematicians, and on the solution of

**differential**equations.The other method starts from the observed values of the periods, and establishes a

**differential**equation from which these periods may be derived.Riecke, 3 who deduced a

**differential**equation of the 10th order.It should be added that the

**differential**treatment of the various classes was by no means confined to the case of wergilds.AdvertisementHis primary aim has been declared to be the advancement and elaboration of the theory of

**differential**equations, and it was with this end in view that he developed his theory of transformation groups, set forth in his Theorie der Transf ormationsgruppen (3 vols., Leipzig, 1888-1893), a work of wide range and great originality, by which probably his name is best known.Monge's memoir just referred to gives the ordinary

**differential**equation of the curves of curvature, and establishes the general theory in a very satisfactory manner; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795.These have often been mistaken for nuclei and have to be carefully distinguished by

**differential**stains.The total power developed is 600 H.P., and though the load factor varies very greatly in this case, the

**differential**type of governor used secures perfect control of the running of the wheels.We thus obtain the

**differential**equation gk(d 2 0/dx 2) =cgdo/dt+hpo, which is satisfied by terms of the type =c" sin where a 2 -b 2 = hp/qk, and ab = urnc/k.The

**differential**equation for the distribution of temperature in this case includes the majority of the methods already considered, and may be stated as follows.There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that his initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the

**differential**calculus.A third form of selection, which may affect the composition of the next generation without of necessity involving a

**differential**death-rate or a**differential**fertility, is assortative mating, or the tendency of those members of one sex which exhibit a particular character to mate only with members of the other sex which exhibit the same or some other definite character.**Differential**fertility may be induced in either of two ways.For an attempt to treat the whole problem of

**differential**fertility and assortative mating numerically, see Pearson, The Grammar of Science, 2nd edition, London, 1900.His well-known correction of Laplace's partial

**differential**equation for the potential was first published in the Bulletin de la societe philomatique (1813).Where accurate

**differential**observations or photographs involving other than instantaneous exposures have to be made, the additional condition is required that the optical axis of the telescope shall accurately and automatically follow the object under observation in spite of the apparent diurnal motion of the heavens, or in some cases even of the apparent motion of the object relative to neighbouring fixed stars.The thermal capacity of the scale-pan, &c., can be determined by a separate experiment., or, still better, eliminated by the

**differential**method of counterpoising with an exactly similar arrangement on the other arm of the balance.If in this we put r= I/u, and eliminate t by means of (15), we obtain the general

**differential**equation of central orbits, viz.Required the relation between the velocity of translation 02 of W and the angular velocity af of the

**differential**barrel.This combination, called Hunters or the

**differential**screw, combines the strength of a large thread with the slowness of motion due to a small one.A conspicuous feature of the New Mexican landscape is the mesa, a flat-topped hill created by

**differential**erosion and projecting above the surrounding country like a table.Electric attractions and repulsions were, however, regarded as

**differential**actions in which the mutual repulsion of the particles of electricity operated, so to speak, in antagonism to the mutual attraction of particles of matter for one another and of particles of electricity for matter.The chief piece of mechanism in the roving frame is the gearing known as the "

**differential**motion."He also developed the theory of

**differential**and of summational tones.The aberrations can also be expressed by means of the "characteristic function " of the system and its

**differential**coefficients, instead of by the radii, &c., of the lenses; these formulae are not immediately applicable, but give, however, the relation between the number of aberrations and the order.Everywhere are evidences of water and wind erosion, of desiccation and

**differential**weathering.The

**differential**diagnosis is of the highest importance, but very frequently time alone will furnish a sufficient criterion.Freedom of navigation of the Congo and all its affluents was also secured, and

**differential**dues on vessels and merchandise were forbidden.Instead of confining himself, as before, to the fruitless integration of three

**differential**equations of the second degree, which are furnished by mathematical principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit.He investigated the problem by means of the general

**differential**equations of static equilibrium for dams of triangular and rectangular form considered as isotropic elastic solids.He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on analytical geometry and the

**differential**calculus.Jacobi and other mathematicians have developed to a great extent, and as a question of pure mathematics only, Hamilton's processes, and have thus made extensive additions to our knowledge of

**differential**equations.Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of

**differential**equations which constantly occur in the treatment of physical questions, only a few items have been published, at intervals, in the Philosophical Magazine.In order to obtain a greater ratio of R to E, without using a large number of sheaves, various arrangements are used, of which the' Weston

**differential**pulley block is a typical example.This new method Baring tried to discover in altering the

**differential**duties on timber and sugar, and substituting a fixed duty of 8s.Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second

**differential**coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).**Differential**equations which express the changes of the co-ordinates are then constructed.In the actual problems of celestial mechanics three co-ordinates necessarily enter, leading to three

**differential**equations and six equations of solution.In a case like the present one, where there are two

**differential**equations of the second order, there will be four such constants.After the establishment of universal gravitation as the primary law of the celestial motions, the problem was reduced to that of integrating the

**differential**equations of the moon's motion, and testing the completeness of the results by comparison with observation.The analytic method sought to express the moon's motion by integrating the

**differential**equations of the dynamical theory.By the second general method the moon's co-ordinates are obtained in terms of the time by the direct integration of the

**differential**equations of motion, retaining as algebraic symbols the values of the various elements.He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "infinitesimals," which he was one of the first to introduce into geometrical demonstrations, was contained the fruitful germ of the

**differential**calculus.And the more advanced Imperialists, as well as the more oldfashioned protectionists (like Mr Chaplin) who formed an integral body of the Conservative party, had looked forward to this tax being converted into a

**differential**one between foreign and colonial corn, so as to introduce a scheme of colonial preference and commercial consolidation between the colonies and the mother country.Leslie's main contributions to physics were made by the help of the "

**differential**thermometer," an instrument whose invention was contested with him by Count Rumford.His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and substitutions, the theory of functions,

**differential**equations and determinants.He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown

**differential**calculus.Relationships where there is a significant power

**differential**are considered dubious.He sent his results on making the

**differential**calculus into a purely algebraic theory to the Royal Society.On the basis of the above analysis, the

**differential**terminal bonus policy had applied since terminal bonus was first introduced in 1973.The DSC 204 Phoenix

**differential**scanning calorimeter can be equipped with an automatic sample changer for up to 64 samples.This has been confirmed by scanning

**differential**calorimetry (DSC ).Surfactant was isolated from duck, chicken, and pig lung lavage fluid by

**differential**centrifugation.Think about the following... Joints, flanges Crevices let stagnant liquid accumulate, causing locally accelerated corrosion from

**differential**aeration.This approach has been proven to be highly resistant against many attacks such as

**differential**and linear cryptanalysis.On closer examination, the algorithms show statistical weaknesses that might allow key recovery using

**differential**cryptanalysis.As always the correct

**differential**diagnosis is necessary before recommending any course of action.Diff lock - A device that locks the diff lock - A device that locks the

**differential**for maximum traction in adverse conditions.And prohibited

**differential**are interactions between doctor outside the.Their ability to maintain a price

**differential**without dramatic losses of market share appears to have reduced.In each case a semantic

**differential**was used from 1 - extremely unlikely to 5 - certain.I've personally never heard of anyone ever having a problem more serious than an oil leak from the rear

**differential**on our cars.At the rear is a limited slip

**differential**with a 3.46 to 1 ratio to reduce rear wheel slip under acceleration.The Japanese version I tried had a number of options, including stiffer sports suspension, a limited-slip

**differential**, heated seats and air-conditioning.Section 2 starts by looking at the direction field associated with a first-order

**differential**equation.We find that religion accounts for about half of the unemployment

**differential**in our two samples.In this approach we use

**differential**calculus and**differential**geometry both to filter and analyze multi-dimensional images.The Numerov method [2] is used to solve the

**differential**equation.These powerful and effective tools are used to solve many problems involving

**differential**equations.In fact, it obeys a

**differential**equation known as the Schrodinger equation (or a generalization thereof ).These systems can be described by a set of ordinary

**differential**equations (ODEs) or map.How

**differential**gene expression leads to organ formation and cellular differentiation.Stochastic

**differential**equations driven by a Wiener process are studied.Thus, in total there are six partial

**differential**equations to be solved.For most of the nonlinear partial

**differential**equations which we have tested, the running time is no more than 10 seconds.Describes care and problems, and includes the etiology, clinical course,

**differential**diagnosis and management of patients.Language goes no further in this direction, but the mathematical symbolism of the

**differential**calculus allows of an indefinite extension of this procedure.Scabies should be considered in the

**differential**diagnosis of any itchy rash affecting the flexures.This would give rise to

**differential**flexure and expansion of the instrument as its various parts cooled at different rates during the following night.This indicates that folic acid can modulate key mechanisms for growth and

**differential**in the central nervous system.**Differential**localized changes in proliferative rates of populations of cells could play a crucial role in the patterning of the developing forebrain.Hamiltonian theory is an important element of integrable systems, whether discrete, ordinary

**differential**or partial**differential**equations.This was especially heartening given the very obvious size

**differential**in the pack.**Differential**methylation of DNA sequences associated with imprinted genes is thought to control genomic imprinting, which occurs soon after fertilization.The only real issue is whether such

**differential**charging would be unduly intrusive into peopleÃ¢s privacy.Areas covered include linear algebra, optimization, quadrature,

**differential**equations, regression analysis, and time series analysis.Such a

**differential**structure, expressing the local versus global nature of brain structure, is lacking in adaptive linear algebra.My Proposal to Improve DAB Synchronous modulation First as I've just described, DAB uses

**differential**modulation.In a third position, justice requires the nullification of some or all effects of

**differential**option luck.The enormous

**differential**is because US stocks are very overpriced and European stocks are somewhat overpriced.His main research interest is the molecular pathology of cancer particularly the

**differential**expressions of proteins.It will pass the test of what you can call

**differential**phenomenology.Programs for newly

**differential**Ln premium to create a. We do our what is best.This

**differential**shrinkage can lead to a shearing action building up between the two materials, resulting in failure.An enclosed vivarium is unsuitable for any species that requires a temperature gradient, or

**differential**, to enable self-selection of body temperature.He invented a

**differential**thermometer, a hygrometer and a photometer, and wrote An Experimental Inquiry into Heat in 1804.Research interests My main research interest is in the application of algebraic topology to

**differential**topology.A

**differential**pressure transducer is placed across the two volumes.There are lessons to be learned from the

**differential**turnout.In seeking a solution to the

**differential**undercount the Census Bureau turned to dual system estimation.A long list of Boole's memoirs and detached papers, both on logical and mathematical topics, will be found in the Catalogue of Scientific Memoirs published by the Royal Society, and in the supplementary volume on

**Differential**Equations, edited by Isaac Todhunter.In the 16th and 17th chapters of the

**Differential**Equations we find, for instance, a lucid account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844.Peacock powerfully aided the movement by publishing in 1820 A Collection of Examples of the Application of the

**Differential**and Integral Calculus.The response to the action of light in diatropic leaves is, according to Haberlandt, due to the presence of epidermal cells which are shaped like a lens, or with lens-shaped thickenings of the cuticle, through which convergence of the light rays takes place and causes a

**differential**illumination of the lining layer of protoplasm on the basal walls of the epidermal cells, by which the stimulus resulting in the orientation of the leaf is brought about.But it was discovered that there were no " indestructible powers of the soil "; that the fertility of land in a country like England is almost entirely the result of improvement at some time or other; that " advantage of situation " includes very much more than the words in their literal sense imply; that both " fertility " and " advantage of situation " include many kinds of

**differential**advantage; that in some circumstances rent does not enter into the price of agricultural and other produce, and that in others it does.An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial

**differential**equations which suffice to determine them, and to ascertain many of their properties.It is amusing to find him speaking jubilantly of the unexpectedly large audience of eight which assembled to hear his first lecture (in 1854) on partial

**differential**equations and their application to physical problems.He had previously published some medical and botanical dissertations, besides his Exercitationes quaedam Mathematicae, containing a solution of the

**differential**equation proposed by Riccati and now known by his name.He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2

**differential**coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see Algebraic Forms).Euler (see

**Differential**Equations).A belted arrangement of relief s and soils, resulting from

**differential**erosion on strata of unlike composition and resistance, characterizes almost the entire area of the coastal plain.In 1874 the university of GÃƒ¶ttingen granted her a degree in absentia, excusing her from the oral examination on account of the remarkable excellence of the three dissertations sent in, one of which, on the theory of partial

**differential**equations, is one of her most remarkable works.It seems right to point out, however, that countervailing duties, which are really

**differential**duties of a special kind, are not the good expedient they are supposed to be for nullifying foreign bounties; that experience of**differential**duties in former times is altogether against them; and that they cannot be enforced without certificates of origin and other causes of harassment and confusion in the conduct of trade.**Differential**refraction pushes the lower limb upwards to create the highly flattened shape.Flow across the resistive element produces a small

**differential**pressure across the two pressure trappings.The results are applied to a class of retarded delay

**differential**equations.Microstructures were studied using a combination of

**differential**scanning calorimetry (DSC) and transmission electron microscopy (TEM).The concept of instability of a basic state is first introduced using models which yield simple ordinary

**differential**equations.Additional features include traction control for an extra £ 395 and a limited slip

**differential**at a cost of £ 995.We show, however, that asymmetric states can be moving solutions of the partial

**differential**equation.**Differential**tax rates become a factor to bear in mind.Detailed analysis may be undertaken in a manner similar to that for the first order ordinary

**differential**equations.**Differential**bone growth between large and small breeds of cattle is usually established prior to a slaughter weight of 500 kg in males.Urine and blood tests may be ordered to check for microscopic amounts of blood in the urine and to obtain a complete

**differential**blood count.The WBC

**differential**helps to distinguish many of these causes.The first screening test for SCID is a white blood cell count with a count of the lymphocytes (

**differential**) because in most forms of SCID the lymphocyte count will be very low.Another

**differential**cause of facial paralysis similar to Bell's palsy is Ramsey-Hunt Syndrome.The symptoms of food poisoning by Shigella organisms may resemble meningitis and a

**differential**diagnosis must be made by isolating the causative bacteria.The

**differential**diagnosis and treatment of posterior plagiocephaly is a challenging aspect of craniofacial surgery.Diagnostic testing starts with a complete blood count (CBC) and

**differential**, counting RBCs, white blood cells (WBCs) and measuring hemoglobin (Hgb), hematocrit (Hct), and other factors.In the microscopic examination of a stained blood smear (

**differential**), red cells may appear smaller than normal.**Differential**diagnosis of botulism can be complex because the symptoms mimic those of other diseases, especially diseases characterized by muscle weakness.**Differential**diagnosis (distinguishing Lyme disease from other diseases) is based on clinical evaluation with laboratory tests used for clarification when necessary.By evaluating a child's developmental profile, a doctor may develop a

**differential**diagnosis.The indices are used to help in the

**differential**diagnosis of anemia.Separation anxiety shares some common features with dependent personality disorder, and should be considered as a

**differential**diagnosis.So is the power

**differential**(of a boss and his direct report) which could have negative consequences for either you and/or your boss.Sexy lingerie for older women has been big business in Europe for years, although there is no

**differential**between the bras, bustiers and garters worn by younger or older women.In the case of the original Repsold plan without clockwork the description is not quite exact, because both the process of following the object and correcting the aim are simultaneously performed; whilst, if the clockwork runs uniformly and the friction-disk is set to the proper distance from the apex of the cone, the star will appear almost perfectly at rest, and the observer has only to apply delicate corrections by

**differential**gear - a condition which is exactly analogous to that of training a modern gun-sight upon a fixed object.The coils of the electromagnets are differentially wound with silk-covered wire, 4 mils (= 004 inch) in diameter, to a total resistance of 400 ohms. This

**differential**winding enables the instrument to be used for " duplex " working, but the connexions of the wires to the terminal screws are such that the relay can be used for ordinary single working.The - - study of tidal strain in the earth's crust by Sir George Darwin has led that physicist to indicate the possibility of the triangular form and southerly direction of the continents being a result of the

**differential**or tidal attraction of the sun and moon.This latter work included the

**differential**and integral calculus, the calculus of variations, the theory of attractions, and analytical mechanics.Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the

**differential**and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "**Differential**Equations," with the topics existence theorems, methods of solution, general theory; "**Differential**Forms and**Differential**Invariants," with the topics**differential**forms, including Pfaffians, transformation of**differential**forms, including tangential (or contact) transformations,**differential**invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the**differential**equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.The theories of determinants and of symmetric functions and of the algebra of

**differential**operations have an important bearing upon this comparatively new branch of mathematics.To Lagrange, perhaps more than to any other, the theory of

**differential**equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear

**differential**equations to definite integrals, and furnished an elegant method by which the linear partial**differential**equation of the second order might be solved.Reference to a geometrical interpretation seems at first sight to throw light on the meaning of a

**differential**coefficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself.The elementary idea of a

**differential**coefficient is useful in reference to the logarithmic and exponential series.By applying the method of the

**differential**calculus, we obtain cos i= { (µ 2 - 1)/(n24-2n)} as the required value; it may be readily shown either geometrically or analytically that this is a minimum.The resolution of the questions concerning the motion of fluids was effected by means of Euler's partial

**differential**coefficients.But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the

**differential**relation at every point = dy dp The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.Both these methods, differing from that now employed, are interesting as preliminary steps towards the method of fluxions and the

**differential**calculus.In the biserial type the polyps on the two sides of the stem have primitively an alternating, zigzag arrangement; but, by a process of

**differential**growth, quickened in the 1st, 3rd, 5th, &c., members of the stem, and retarded in the 2nd, 4th, 6th, &c., members, the polyps may assume secondarily positions opposite to one another on the two sides of the stem.For the subjects of this heading see the articles

**DIFFERENTIAL**EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the

**differential**calculus by Newton and Leibnitz and the discovery of gravitation.Between them the general theory of the complex variable, and of the various "infinite" processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the

**differential**geometry of surfaces.From the

**differential**coefficients of the y's with regard to the x's we form the functional.The Partial

**Differential**Equations.--It will be shown later that covariants may be studied by restricting attention to the leading coefficient, viz.Solving the equation by the Ordinary Theory Of Linear Partial

**Differential**Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.Laplace owned that he had despaired of effecting the integration of the

**differential**equations relative to secular inequalities until Lagrange showed him the way.Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many equations as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the

**differential**equations of motion thence deduced by simple differentiation.He regulated and simplified the whole system of taxation, encouraged agriculture by

**differential**duties in favour of the farmers, and promoted trade by a systematic improvement of the ways of communication.The well-known Treatise on

**Differential**Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his

**Differential**Equations much more complete than the first edition; and part of his last vacation was spent in the libraries of the Royal Society and the British Museum.The two methods most commonly employed are the

**differential**and bridge methods.Since dp4+(-)P+T1(p +q qi 1)!dd4, the solutions of the partial

**differential**equation d P4 =o are the single bipart forms, omitting s P4, and we have seen that the solutions of p4 = o are those monomial functions in which the part pq is absent.He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial

**differential**equations, and Jacobi's on analytical mechanics and higher algebra.In expounding the principles of the

**differential**calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions.The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the

**differential**and integral calculus of an analogous method depending wholly on the serial development of algebraical functions.At the end of 1889 Crispi abolished the

**differential**duties against French imports and returned to the general Italian tariff, but France declined to follow his lead and maintained her prohibitive dues.It has been shown above that a covariant, in general, satisfies four partial

**differential**equations.By considering only the particles of air found in a right line, he reduced the problem of the propagation of sound to the solution of the same partial

**differential**equations that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject.He proves, by means of the six linear partial

**differential**equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.