Joseph-Louis Lagrange

How to use this course

Physicists meet Lagrange in the Lagrangian, the function whose stationary integral gives the laws of motion. Engineers meet him in Lagrange multipliers, astronomers in the Lagrange points, algebraists in Lagrange’s theorem on subgroups, number theorists in the four-square theorem, and students of calculus in the Lagrange form of the remainder. These are not scattered results. Lagrange’s lifelong aim was to replace geometric ingenuity with general analytic methods, and nearly everything he touched became a method.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Lessons 04 to 07 form the mechanics strand, Lessons 08 and 09 the algebra and arithmetic strand, and Lesson 10 his attempt to found calculus on power series.

Timeline

Turin, 1736–1755

Giuseppe Lodovico Lagrangia was born in Turin on 25 January 1736. His father, Giuseppe Francesco Lodovico Lagrangia, was treasurer of the Office of Public Works and Fortifications; his mother, Teresa Grosso, was the only daughter of a physician from Cambiano near Turin. The family had French ancestry on the father’s side, and the young man often signed himself Lagrange or de la Grange. He was the eldest of eleven children, only two of whom reached adulthood. The family’s fortune was lost in poor speculation, a loss he later said was lucky, since wealth might have kept him from mathematics.

His father intended him for the law, and he studied at the College of Turin. Geometry in the style of Euclid left him cold. What caught him, at about seventeen, was an essay by Edmond Halley from 1693 on the advantages of algebra in optics. He taught himself analysis from the books available, quickly and alone. Giovanni Battista Beccaria’s teaching drew him towards physics.

A first paper and a mistake

On 23 July 1754 he published his first work, a letter in Italian to Giulio Fagnano, under the name Luigi De la Grange Tournier. It set out an analogy between the binomial theorem and the successive derivatives of a product. He then learned that Leibniz and Johann Bernoulli had found the same thing in their correspondence half a century earlier. The discovery embarrassed him and made him determined to do something original.

The artillery school and the Turin Academy, 1755–1766

Within a year he found it. Working on the tautochrone, the curve along which a sliding bead reaches the bottom in the same time from any starting point, he devised a purely analytic method for problems of maxima and minima of integrals. On 12 August 1755 he described it in a letter to Leonhard Euler in Berlin, the leading authority on such questions. Euler replied on 6 September with warm praise. He had been working on the same problems with geometric methods, and he adopted Lagrange’s approach, gave the subject the name calculus of variations, and generously acknowledged the younger man’s method as superior to his own. Lesson 04 explains the method.

On 28 September 1755, aged 19, Lagrange was appointed professor of mathematics at the Royal Artillery School in Turin. On 2 September 1756, on Euler’s proposal, he was elected to the Berlin Academy. In 1757 he joined two friends in founding a private scientific society, which became the Royal Academy of Sciences of Turin. Its journal, the Miscellanea Taurinensia, carried most of his early work: the calculus of variations, the propagation of sound and vibrating strings (Lesson 06), and the integration of differential equations.

Prizes

The Paris Académie des Sciences set prize questions in celestial mechanics. Lagrange won in 1764 with an essay on the libration of the Moon, explaining why the Moon always turns nearly the same face to the Earth, and in 1766 with an essay on the satellites of Jupiter. Jean le Rond d’Alembert, who met him on a visit to Paris in 1763–64, became a friend and an energetic promoter. Lagrange was underused and isolated in Turin, and he refused to be his own advocate at court. He was also often unwell and worked to exhaustion.

Berlin, 1766–1787

In 1766 Euler left Berlin for Saint Petersburg. D’Alembert persuaded Frederick II of Prussia to offer the post to Lagrange; Frederick wrote that the greatest mathematician in Europe should be at the court of the greatest king. Lagrange left Turin in August, arrived in Berlin in October, and on 6 November 1766 succeeded Euler as director of mathematics at the Berlin Academy. He stayed twenty years.

The Berlin years were the most productive of his life. He had no teaching duties and was paid to write memoirs for the Academy. In 1767 he married his cousin Vittoria Conti, who had come from Turin to keep house for him. They had no children. She was often ill, and he nursed her through a long decline until her death in 1783, a loss that left him deeply depressed.

The work of Berlin

A brief list shows the range. In number theory: Pell’s equation, the four-square theorem (1770), Wilson’s theorem (1771) and the theory of quadratic forms (1773–75) (Lesson 09). In algebra: the Réflexions sur la résolution algébrique des équations (1770–71), which analysed why the classical formulas work and why they might fail for the quintic (Lesson 08). In celestial mechanics: the three-body problem (1772), the secular theory of the planets with Pierre-Simon Laplace, and Paris prizes in 1772 (shared with Euler), 1774 and 1780 (Lesson 07). In mechanics: the plan of a treatise built entirely on analysis, which he described to Laplace in 1782 (Lesson 05).

Frederick II died in 1786, and the climate in Berlin cooled towards foreign academicians. Lagrange received several invitations and accepted one from Paris, on terms that excused him from teaching. On 18 May 1787 he left Berlin.

The calculus of variations

For the general reader. Which curve gives the fastest slide between two points? Which surface spanning a wire loop has the least area? These questions ask for a whole function that makes some quantity smallest. At nineteen Lagrange found a general recipe: vary the unknown function slightly, demand that the first-order change vanish, and read off a differential equation the function must satisfy.

The \(\delta\) method

Consider the functional

over twice continuously differentiable functions \(y\) with fixed values \(y(a),y(b)\). Lagrange introduced a symbol \(\delta y\) for an arbitrary small variation of the whole function, vanishing at the endpoints, with \(\delta y'=(\delta y)'\). The first variation is

after integrating by parts. The boundary term vanishes. The fundamental lemma of the calculus of variations (if \(\int g\,\eta=0\) for all smooth \(\eta\) vanishing at the ends and \(g\) is continuous, then \(g\equiv0\)) yields:

Euler had obtained the equation in 1744 by discretizing curves into polygons. Lagrange’s derivation worked directly with variations of functions and extended at once to several unknown functions, higher derivatives, multiple integrals and variable endpoints, which yield natural boundary conditions \(F_{y'}=0\) where the endpoint is free.

Minimal surfaces

In the 1760–61 Miscellanea Taurinensia Lagrange applied the method to a surface \(z=f(x,y)\) of least area, minimizing \(\iint\sqrt{1+f_x^2+f_y^2}\,dx\,dy\). He obtained the minimal surface equation

which Jean-Baptiste Meusnier later interpreted as the vanishing of the mean curvature.

Constraints

To extremize \(J\) subject to an integral constraint \(K[y]=\int G\,dx=c\), one extremizes \(J-\lambda K\) without constraint, with a constant \(\lambda\) fixed by the condition \(K=c\). For a pointwise constraint \(g(x,y,y')=0\) the multiplier becomes a function \(\lambda(x)\). This is the variational form of the multiplier rule of Lesson 05.

Analytical mechanics

For the general reader. Newton’s mechanics adds up forces as arrows, which becomes painful for a pendulum on a moving cart or a spinning top. Lagrange rewrote mechanics so that one picks any convenient coordinates, writes down two scalar functions, the kinetic and potential energies, and turns a handle. His 1788 book famously contains no diagrams at all.

From virtual work to Lagrange’s equations

For particles of masses \(m_i\) at positions \(\mathbf r_i\), subject to ideal constraints that do no work in admissible displacements, d’Alembert’s principle states

for all virtual displacements \(\delta\mathbf r_i\) compatible with the constraints. Lagrange made this the single axiom of the Mécanique analytique. If the configuration is described by independent generalized coordinates \(q_1,\dots,q_n\), so that \(\mathbf r_i=\mathbf r_i(q,t)\), the identities \(\partial\dot{\mathbf r}_i/\partial\dot q_k=\partial\mathbf r_i/\partial q_k\) and \(\frac{d}{dt}\frac{\partial\mathbf r_i}{\partial q_k}=\frac{\partial\dot{\mathbf r}_i}{\partial q_k}\) turn the inertial terms into derivatives of the kinetic energy \(T=\sum\tfrac12m_i|\dot{\mathbf r}_i|^2\).

The second form is the Euler–Lagrange equation of Lesson 04 for the action \(S=\int L\,dt\). This is Hamilton’s principle, stated in this form by William Rowan Hamilton in 1834–35: the motion makes the action stationary among paths with fixed endpoints.

Multipliers

When the coordinates are not independent, Lagrange kept them and adjoined the constraints with undetermined multipliers. In statics this gives the method taught in every calculus course.

The proof uses the implicit function theorem: near \(x^*\) the constraint set is a smooth manifold whose tangent space is the common kernel of the \(\nabla g_j\). Since \(\nabla f(x^*)\) annihilates this kernel, it lies in the span of the \(\nabla g_j\).

Small oscillations, sound and fluids

For the general reader. A system shaken a little about a stable rest position moves as a superposition of independent “normal modes”, each with its own frequency. Lagrange made this a general theorem, starting from the vibrating string and the propagation of sound. He also introduced a way of describing fluids by following individual particles, and proved that a whirl-free flow of an ideal fluid stays whirl-free.

Normal modes

Near a stable equilibrium \(q=0\) of a system with \(L=T-V\), expand \(T\approx\tfrac12\dot q^{\mathsf T}M\dot q\) and \(V\approx\tfrac12q^{\mathsf T}Kq\), with \(M\) positive definite and \(K\) symmetric. The linearized equations are \(M\ddot q+Kq=0\).

The proof is simultaneous diagonalization of two quadratic forms, one positive definite. The determinant equation is still sometimes called the secular equation, because Lagrange and Laplace met it in the secular theory of the planets. Lagrange assumed the roots were distinct; Karl Weierstrass showed in 1858 that repeated roots cause no secular growth when \(M\) is positive definite.

The vibrating string

In 1759 Lagrange entered the dispute between d’Alembert, Euler and Daniel Bernoulli over the vibrating string. He treated a massless string carrying \(n\) equal beads, solved the finite system exactly by normal modes, and let \(n\to\infty\). For \(n\) beads of mass \(m\) spaced \(h\) apart under tension \(\tau\), with fixed ends, the mode frequencies are

As \(n\to\infty\) with \(L=(n+1)h\) and density \(\mu=m/h\) fixed, \(\omega_k\to\frac{k\pi}{L}\sqrt{\tau/\mu}\), the harmonics of the continuous string. His limit procedure came close to the Fourier sine series, without his drawing that conclusion.

Fluids

The Mécanique analytique treats hydrostatics and hydrodynamics by the same principle as particles. In the Lagrangian description each fluid particle is labelled by its initial position \(\mathbf a\), and the motion is a map \(\mathbf x(\mathbf a,t)\); the Eulerian description gives the velocity field \(\mathbf v(\mathbf x,t)\) instead. Both descriptions appear already in Euler’s work, but the names stuck.

A modern proof uses the vorticity equation \(\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf v-\boldsymbol\omega(\nabla\cdot\mathbf v)\), which is linear in \(\boldsymbol\omega\) along particle paths, so \(\boldsymbol\omega=0\) initially implies \(\boldsymbol\omega\equiv0\). Lagrange’s own argument assumed analyticity in time; Augustin-Louis Cauchy gave a sharper form in 1815, and Hermann von Helmholtz generalized it to vortex lines in 1858.

The three-body problem and perturbations

For the general reader. Two bodies under gravity follow ellipses. Three bodies have no general solution in closed form. Lagrange found exceptional configurations where three bodies keep the same shape forever: in a line, or at the corners of an equilateral triangle. In 1906 an asteroid was found sharing Jupiter’s orbit at one of the triangle points, and space telescopes now park at these points.

The 1772 essay

In his Essai sur le problème des trois corps, which shared the Paris prize of 1772 with Euler, Lagrange reduced the general three-body problem to equations for the mutual distances. He found two families of solutions in which the triangle formed by the bodies keeps its shape while rotating and possibly changing size. Euler had found the collinear ones, published in 1765; the equilateral ones were new.

Derivation. Put the centre of mass at the origin and take \(|\mathbf r_i-\mathbf r_j|=d\) for all pairs. The force on body 1 is \(\frac{Gm_1}{d^3}\big(m_2(\mathbf r_2-\mathbf r_1)+m_3(\mathbf r_3-\mathbf r_1)\big)\). Since \(m_1\mathbf r_1+m_2\mathbf r_2+m_3\mathbf r_3=0\), this equals \(-\frac{Gm_1(m_1+m_2+m_3)}{d^3}\mathbf r_1\): a central force towards the centre of mass, proportional to distance. The same holds for each body, so rigid rotation with \(\Omega^2=G\sum m/d^3\) solves the equations.

The restricted problem

Let two masses \(1-\mu\) and \(\mu\) move in circles, and use units in which their separation, \(G\) times the total mass, and \(\Omega\) are all 1. In the rotating frame they sit at \((-\mu,0)\) and \((1-\mu,0)\), and a massless particle moves under the effective potential

The equilibria are the critical points of \(U\): three on the axis (\(L_1,L_2,L_3\)) and two at \(r_1=r_2=1\) (\(L_4,L_5\)). The integral \(C=2U-\dot x^2-\dot y^2\), found by Carl Gustav Jacobi, bounds the regions the particle can reach.

Linearizing at \(L_4\) gives the characteristic equation \(\lambda^4+\lambda^2+\tfrac{27}{4}\mu(1-\mu)=0\), whose roots are purely imaginary exactly when \(1-27\mu(1-\mu)>0\). The Sun–Jupiter system (\(\mu\approx0.001\)) qualifies, and thousands of Trojan asteroids, the first being 588 Achilles, found by Max Wolf in 1906, librate about Jupiter’s \(L_4\) and \(L_5\).

Variation of the elements

Lagrange developed the method of variation of constants into a general tool. For a planet with disturbing function \(\mathcal R\), he expressed the rates of change of the six elements as linear combinations of the partial derivatives of \(\mathcal R\), with coefficients given by what are now called Lagrange brackets of the elements. Two of the resulting Lagrange planetary equations are

With Laplace he built on them the secular theory of the planets; in 1776 he proved the invariance of the mean distances at first order in the masses without expanding in the eccentricities.

The algebraic resolution of equations

For the general reader. Formulas for solving equations of degree 2, 3 and 4 had been known since the sixteenth century, but nobody had found one for degree 5. Lagrange asked why the known formulas work. His answer was that they depend on how certain expressions in the roots change when the roots are shuffled. That question about shuffles led Paolo Ruffini, Niels Henrik Abel and Évariste Galois to show that no general formula exists for degree 5, and it is the seed of group theory.

Resolvents

In the Réflexions sur la résolution algébrique des équations (1770–71) Lagrange reviewed every known method and found a common pattern. For a cubic with roots \(x_1,x_2,x_3\) and \(\omega=e^{2\pi\mathrm i/3}\), set

Under the six permutations of the roots, \(t\) takes six values, but \(t^3\) takes only two, and these two are interchanged by transpositions. So \(t^3\) satisfies a quadratic whose coefficients are symmetric in the roots, hence known from the coefficients of the cubic. Solving the quadratic, then taking cube roots, gives \(t\) and \(t'=x_1+\omega^2x_2+\omega x_3\), and then

with similar formulas for \(x_2,x_3\). The quartic works the same way with the expression \(x_1x_2+x_3x_4\), which takes three values under permutation and so satisfies a cubic, the resolvent cubic.

Counting values

The proof groups permutations into cosets \(\sigma H\): two permutations give the same function exactly when they lie in the same coset, and each coset has \(|H|\) elements. The same argument for any finite group \(G\) and subgroup \(H\) gives Lagrange’s theorem \(|G|=[G:H]\,|H|\), named in his honour though he never had the concept of a group.

The quintic

For degree 5 Lagrange looked for a function of the roots taking fewer than 5 values that would reduce the problem to lower degree, and found the search unpromising. He concluded that the general solution of the quintic would need a different idea, if it was possible at all. Ruffini (1799, with gaps) and Abel (1824) proved it impossible; Galois explained in terms of the group of the equation which equations are solvable.

Number theory

For the general reader. Pierre de Fermat had left a list of claims about whole numbers without proofs, and Euler had spent decades proving some of them. Lagrange, in a few years in Berlin, proved several that had resisted Euler: that every whole number is a sum of four squares, and that Pell’s equation always has solutions. He also began the systematic theory of quadratic forms that Gauss completed.

Four squares

Proof sketch. Euler’s identity shows that a product of two sums of four squares is a sum of four squares:

So it suffices to treat primes \(p\). For odd \(p\), the \((p+1)/2\) values \(x^2\) and the \((p+1)/2\) values \(-1-y^2\) modulo \(p\) cannot be disjoint, so \(x^2+y^2+1\equiv0\pmod p\) for some \(x,y\): some multiple \(mp\) with \(0<m<p\) is a sum of four squares. Lagrange’s descent then shows that if \(m>1\) is the least such multiple, a smaller one exists, a contradiction. Hence \(m=1\).

Pell’s equation and continued fractions

The fundamental solution appears among the convergents of the continued fraction of \(\sqrt d\), whose periodicity Lagrange proved by showing that the complete quotients belong to a finite set of reduced quadratic irrationals. William Brouncker and others had given methods to find solutions; Lagrange gave the first complete proof that the method always succeeds.

Quadratic forms and Wilson’s theorem

In the Recherches d’arithmétique (1773–75) Lagrange called two binary quadratic forms \(ax^2+bxy+cy^2\) equivalent if an integer change of variables of determinant \(\pm1\) carries one to the other. He proved that every form of discriminant \(D=b^2-4ac<0\) is equivalent to a reduced one with \(|b|\le a\le c\), so there are finitely many classes for each \(D\). Carl Friedrich Gauss built the Disquisitiones arithmeticae (1801) on this foundation. Lagrange also proved, in 1771, Wilson’s theorem: \((p-1)!\equiv-1\pmod p\) for every prime \(p\). The key step is his theorem that a polynomial of degree \(n\) has at most \(n\) roots modulo a prime.

Analytic functions, remainders, interpolation

For the general reader. In the eighteenth century calculus rested on infinitely small quantities that nobody could define. Lagrange tried to rebuild it on algebra alone, by defining derivatives as coefficients of power series. The foundation failed, but it gave us the notation \(f'(x)\), the word “derivative”, and the precise error formula for Taylor polynomials.

The programme of 1797

The Théorie des fonctions analytiques (1797), based on his École Polytechnique lectures, asserted that any function can be expanded as

and defined the fonction dérivée \(f'(x)\) as the coefficient \(p\), with \(f''\), \(f'''\) defined by iteration. This avoided infinitesimals, but the assumption is false: Cauchy observed in 1823 that \(e^{-1/x^2}\) (extended by 0 at 0) has all derivatives zero at 0, so its Taylor series does not represent it. Cauchy then founded calculus on limits instead.

The remainder

Proof. Fix \(x\ne a\), define \(M\) by requiring the formula to hold with \(M\) in place of \(f^{(n+1)}(\xi)/(n+1)!\), and set \(g(t)=f(t)-\sum_{k\le n}\frac{f^{(k)}(a)}{k!}(t-a)^k-M(t-a)^{n+1}\). Then \(g(a)=g'(a)=\cdots=g^{(n)}(a)=0\) and \(g(x)=0\). Rolle’s theorem applied repeatedly gives \(\xi\) with \(g^{(n+1)}(\xi)=0\), that is \(f^{(n+1)}(\xi)=(n+1)!\,M\). Lagrange’s own argument bounded the remainder between the extreme values of \(f^{(n+1)}\) and used the intermediate value property; the case \(n=0\) is the mean value theorem.

Interpolation

Given distinct nodes \(x_0,\dots,x_n\) and values \(y_0,\dots,y_n\), the unique polynomial of degree at most \(n\) through the points is

Edward Waring published the formula in 1779 and Euler a related one in 1783; Lagrange gave it in his École Normale lectures of 1795, and it bears his name.

Paris and the Revolution, 1787–1813

Lagrange arrived in Paris in 1787 as a pensionnaire of the Académie des Sciences, with lodgings in the Louvre. The Mécanique analytique appeared in 1788. Its preface announced that the reader would find no figures in it, only algebraic operations, and that mechanics would become a branch of analysis. He seems to have been in a long depression in these years and did little new research.

The Revolution

When the Revolution came he stayed. In May 1790 he joined the Académie’s commission on weights and measures. In 1792 he married Renée-Françoise-Adélaïde Le Monnier, the young daughter of his astronomer colleague Pierre Charles Le Monnier, and the marriage restored his spirits. In September 1793 a decree ordered the arrest of foreigners born in enemy countries; Lagrange was exempted by name, through Lavoisier’s intervention. When the Académie was suppressed in August 1793, the commission on weights and measures continued, with Lagrange as its chairman. He favoured a strictly decimal system, and the metre and kilogram with decimal subdivisions were adopted. On the execution of Antoine Lavoisier in May 1794 he remarked that it took a moment to cut off that head, and a hundred years might not produce another like it.

Teaching

The École Polytechnique was founded in March 1794, and Lagrange became its first professor of analysis. He also taught at the École Normale in 1795, where Joseph Fourier heard him and judged him a poor speaker, with a strong accent, but a profound teacher. His lectures became the Théorie des fonctions analytiques (1797) and the Leçons sur le calcul des fonctions. He was a founding member of the Bureau des Longitudes in 1795.

Under Napoleon

Napoleon Bonaparte admired him. Lagrange was appointed to the Senate in December 1799, made a grand officer of the Legion of Honour, and created a count of the Empire in 1808. He spent his last years revising the Mécanique analytique, whose first revised volume appeared in 1811.

Death and legacy

In early 1813 his health failed. On 3 April Napoleon awarded him the Grand Cross of the Order of the Reunion. He died in Paris on 10 April 1813, aged 77, and was buried in the Panthéon. The second volume of the revised Mécanique analytique appeared posthumously in 1815. His name is one of the 72 on the Eiffel Tower.

Legacy

• Mechanics and physics: generalized coordinates, the Lagrangian and Lagrange’s equations are the starting point of Hamiltonian mechanics, of field theory and of quantum mechanics through the path integral. The Lagrangian density of the Standard Model is a direct descendant.
• Variational calculus and optimization: the Euler–Lagrange equation, natural boundary conditions and the multiplier rule, from optimal control to the Karush–Kuhn–Tucker conditions of modern optimization.
• Celestial mechanics: the Lagrange points, the planetary equations and, with Laplace, the secular theory of the solar system. Spacecraft such as SOHO and the James Webb Space Telescope operate near the Sun–Earth \(L_1\) and \(L_2\) points.
• Algebra: resolvents and the analysis of permutations of roots, the source of the work of Ruffini, Abel, Galois and Cauchy on permutations, and of Lagrange’s theorem in group theory.
• Number theory: the four-square theorem, continued fractions and Pell’s equation, and the reduction of quadratic forms, the foundation for Gauss.
• Analysis: the notation \(f'\), the mean value theorem and the remainder in Taylor’s theorem, and interpolation.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Joseph-Louis Lagrange, MacTutor History of Mathematics.
2. Joseph-Louis Lagrange, Wikipedia; Joseph-Louis Lagrange, Wikipédia (French).
3. J.-B. Delambre, Notice sur la vie et les ouvrages de M. le comte J.-L. Lagrange, in Œuvres de Lagrange, vol. 1, Gauthier-Villars, 1867.
4. C. G. Fraser, Joseph Louis Lagrange’s algebraic vision of the calculus, Historia Mathematica 14 (1987).
5. J. L. Lagrange, Œuvres de Lagrange, ed. J.-A. Serret and G. Darboux, 14 vols., Gauthier-Villars, 1867–92.

Mathematics and physics

1. J. L. Lagrange, Mécanique analytique, Paris, 1788; English translation Analytical Mechanics by A. Boissonnade and V. N. Vagliente, Kluwer, 1997.
2. J. L. Lagrange, Réflexions sur la résolution algébrique des équations, Nouv. Mém. Acad. Berlin (1770–71).
3. J. L. Lagrange, Essai sur le problème des trois corps, Prix de l’Académie des Sciences (1772).
4. J. L. Lagrange, Théorie des fonctions analytiques, Paris, 1797.
5. C. Lanczos, The Variational Principles of Mechanics, University of Toronto Press, 1949 (Dover reprint).
6. H. M. Edwards, Galois Theory, Springer, 1984 (Lagrange’s resolvents in historical context).
7. A. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, Birkhäuser, 1984.
8. C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999.
9. Lagrange point and Lagrange polynomial, Wikipedia.
10. J. Laskar, Is the solar system stable?, arXiv:1209.5996 (history of the secular theory).