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5. Further Development of the Paradigm: 1858–1874

verfasst von : Thomas Hawkins

Erschienen in: The Mathematics of Frobenius in Context

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Abstract

In his paper of 1858, Weierstrass had considered pairs of real quadratic forms $\Phi = {x}^{t}Bx$, $\Psi = {x}^{t}Ax$, with Φ definite, because his goal was to establish a generalization of the principal axes theorem for them that would provide the basis for a nongeneric treatment of $B\ddot{y} = Ay$.

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Vorheriges Kapitel The Paradigm: Weierstrass’ Memoir of 1858

Nächstes Kapitel The Problem of Pfaff

For example, let P be any 3 ×3 orthogonal matrix, and take $A = PD_{1}{P}^{t}$, B = P D ₂ P ^t, where $D_{1} =\mathrm{ Diag.\,Matrix}(1, 1,-1)$ and $D_{2} =\mathrm{ Diag.\,Matrix}(1, 1,-3)$. Then A and B are symmetric and indefinite. Obviously, since ${P}^{t} = {P}^{-1}$, we have ${P}^{t}AP = D_{1}$ and P ^t B P = D ₂, even though $f(u,v) =\det (uA - vB) = {(u - 2v)}^{2}(u - 3v)$ has (2, 1) as a double root.

If $B ={\Biggl ( \begin{array}{lll} 1\! & \!1\! & \!0 \\ 1\! & \!2\! & \!1 \\ 0\! & \!1\! & \!0 \end{array} \Biggr )}$ and $A ={\Biggl ( \begin{array}{lll} 1\! & \!1\! & \!1 \\ 1\! & \!1\! & \!1 \\ 1\! & \!1\! & \!1 \end{array} \Biggr )}$, then $f(s) =\det (sB - A) = -{s}^{3}$, so s = 0 is a root of multiplicity m = 3, but s ² does not divide all minors of s B − A, e.g., ${s}^{2} - f_{11}(s) = -s(1 - s)$.

It is easily seen that for all i < m, one has $\partial u_{m}/\partial x_{i} = \partial v_{m}/\partial y_{i} = 0$. For example, $\partial u_{m}/\partial x_{i} =\sum _{\sigma \in S_{m}}\mathrm{sgn}\,(\sigma )a_{1,\sigma (1)}\cdots a_{i,\sigma (i)}\cdots a_{i,\sigma (m)} = 0$, since it represents the determinant of a matrix with ith and mth rows equal.

The proof started as follows. If λ is a characteristic root of $C = A + iB$, then $z = p + iq\not =0$ exists such that $\lambda z = Cz = (A + iB)z$. From this, Clebsch concluded without any proof [95, p. 327, eqn. (14)] that

$$\displaystyle{ \lambda p = Ap - Bq\quad \mathrm{and}\quad \lambda q = Aq + Bq. }$$

(5.6)

These equalities follow readily by taking real and imaginary parts if λ, p, and q are assumed real, but Clebsch assumed that (5.6) held without these reality assumptions. I have been unable to see how his assumption can be justified. Using (5.6), Clebsch correctly proved by what would now be translated into inner product considerations that λ must be real [95, pp. 327–329].

Although he may have been encouraged by the precedent of Clebsch’s proof, Christoffel’s actual proof seems to owe more to observations made by Lagrange, as indicated below.

Clebsch began his proof by considering any two characteristic roots λ, λ ′ of B. His proof follows directly from the fact that nonzero $e$ and $e^{\prime}$ exist for which $Be =\lambda e$ and $Be^{\prime} =\lambda e^{\prime}$. He observed that these equations imply that

$$\displaystyle{ \lambda \sum _{j}e_{j}e_{j}^{\prime} =\sum _{jk}b_{jk}e^{\prime}_{j}e_{k}\quad \mathrm{and}\quad \lambda ^{\prime}\sum _{j}e_{j}e_{j}^{\prime} =\sum _{jk}b_{jk}e_{j}e^{\prime}_{k}, }$$

(5.7)

where the e _j and e ′ _j are the respective components of $e$ and $e^{\prime}$. In more familiar notation, the first equation in (5.7) is $\lambda (e \cdot e^{\prime}) = (Be \cdot e^{\prime})$, and the second is $\lambda (e \cdot e^{\prime}) = (e \cdot \lambda ^{\prime}e^{\prime}) = (e \cdot Be^{\prime})$, where $(e \cdot f) ={ \mathrm{e}}^{t}f$ is the usual real inner product. Adding these two equations and invoking the skew-symmetry of B, he concluded that

$$\displaystyle{ (\lambda +\lambda ^{\prime})(e \cdot e^{\prime}) = 0. }$$

(5.8)

That is, he realized that by virtue of skew-symmetry, the right-hand sides of the equations in (5.7) are negatives of one another. Indeed, in matrix notation the right-hand side of the first equation is $Be \cdot e^{\prime}$, which is the same as $e \cdot {B}^{t}e^{\prime} = -(e \cdot Be^{\prime})$, the negative of the right hand side of the second equation. To complete the proof, Clebsch took $\lambda ^{\prime} =\bar{\lambda }$. Since B is real, taking conjugates in $Be =\lambda e$ yields $B\bar{e} =\bar{\lambda }\bar{ e}$, and so he took $e^{\prime} =\bar{ e}$. In this case (5.8) becomes $(\lambda +\bar{\lambda })(e \cdot \bar{ e}) = 0$, whence $\lambda +\bar{\lambda } = 0$ and λ is a pure imaginary number.

For biographical details and documents concerning Christoffel and his work see [249, 319].

Geiser [249, p. vi] suggests that this personality trait also developed during the solitary years 1856–1859.

In a letter of 24 March 1885 to Sonya Kovalevskaya Weierstrass referred to Christoffel as a “wunderlicher Kauz” [441, pp. 194–195]. The letter is also contained in [28].

That is, given a bilinear form $F = {x}^{t}Ay$, a nonsingular linear transformation $x = Px^{\prime}$, $y = Py^{\prime}$ induces a transformation T _P: A → A ′ of the coefficients of F, namely what we can now write as T _P(A) = P ^t A P. An example of a (relative) invariant is $I_{\lambda,\mu }(A) =\det (\lambda A +\mu {A}^{t})$ for any fixed values of λ, μ, since by Cauchy’s product theorem, $I_{\lambda,\mu }(A^{\prime}) = {(\det P)}^{2}I_{\lambda,\mu }(A)$. As we shall see in the next section, Kronecker introduced the determinant I _λ, μ(A) in 1866, as Christoffel realized.

At the same session, Weierstrass summarized one of his papers. See p. 612 of Monatsberichte der Akademie der Wiss. zu Berlin 1866 (Berlin, 1867).

Although Kronecker moved to Berlin as an independent scholar in 1855, it was not until 1861 that he became a member of the Berlin Academy. Thus when Weierstrass presented his paper [587] to the academy, Kronecker was probably not in attendance.

Simple examples with D(u, v) ≡ 0 with P, Q ≠ 0 are given by $P ={\biggl ( \begin{array}{cc} {a}^{2}\! & \!ab \\ ab\! & \!{b}^{2} \\ \end{array} \biggr )}$ and $Q ={\biggl ( \begin{array}{cc} ac\! & \!bc \\ ad\! &\!bd \\ \end{array} \biggr )}$.

Weierstrass explained in a footnote added to the version of his paper that appeared in his collected works (Werke 2 (1902), p. 19n) that “This case [D(u, v) ≡ 0] was not treated by me because I knew that Mr. Kronecker would investigate it thoroughly. (See the relevant works of Mr. K in the monthly reports [Monatsberichten] of the academy.)” The “relevant works” of Kronecker are considered in the next section.

“Hr. Kronecker knüpfte an den versehenden Vortrag folgende Bemerkungen an: …,” Monatsberichte der Akademie der Wiss. zu Berlin 1868, pp. 339–346. Reprinted in Kronecker’s collected works with the title “Über Schaaren quadratischer Formen” [354].

The footnote occurs on the first page of [588] but is omitted from the edited version that Weierstrass included in his collected works.

Unbeknownst to Weierstrass, arithmetic analogues of the W-series and invariant factors had been introduced a few years earlier (1861) by H.J.S. Smith. His work and its relation to Frobenius’ rational version of Weierstrass’ theory of elementary divisors are discussed in Chapter 8.

If $B_{e_{i}}(a_{i})$ denotes the Jordan block corresponding to $W_{e_{i}}(a_{i})$, then $J_{e_{i}}P_{e_{i}} = I_{e_{i}}$ and $W_{e_{i}}(a_{i})P_{e_{i}} = B_{e_{i}}(a_{i})$, where $P_{e_{i}}$ is the permutation matrix corresponding to the permutation

$$\displaystyle{\left (\begin{array}{cccc} 1 & 2 & \cdots &e_{i} \\ e_{i}&e_{i} - 1 & \cdots & 1\\ \end{array} \right ).}$$

Thus if $P = P_{e_{1}} \oplus \cdots \oplus P_{e_{r}}$, then the nonsingular transformations $X = X^{\prime}$, $Y = PY ^{\prime}$ take Weierstrass’ canonical form $s\tilde{\Phi } -\tilde{ \Psi }$ into the bilinear form ${(X^{\prime})}^{t}(sI_{n} - J)Y ^{\prime}$, where $J = B_{e_{1}}(a_{1}) \oplus \cdots \oplus B_{e_{r}}(a_{r})$ is the Jordan canonical form.

The reasoning that led Weierstrass to his formulas for H and K was based on an assumption that was later seen to be far from obvious when s A − B is symmetric, so that the conclusion that H = K when the forms are quadratic—and so the proof of Corollary 5.10—was seen to contain a gap. On the efforts to rework Weierstrass’ theory so that Corollary 5.10 followed, including the important role played by Frobenius, see Section 16.1.

Nineteenth-century mathematicians developed Galois’ work as it was known through his collected works as published posthumously in the Journal de mathématiques pures et appliquées in 1846 [239].

The same result on the integration of $A\,dx/dt = Bx$ was communicated by Weierstrass to the Berlin Academy in 1875, but apparently first published in his Werke [591]. No reference was made to Jordan’s note [323].

For those interested in following it, here is a list of the sources in chronological order: Jordan [324], Kronecker [356, 357], Jordan (before seeing [357]) [325], Kronecker [358, 360], Jordan [326].

The term “rank” was introduced by Frobenius in 1877, but the notion (without a name) was in existence much earlier. See Section 6.3.

If $\mathcal{Q}_{1} = {x}^{t}(uA + vB)x$ and $\mathcal{Q}_{2} = {X}^{t}(u\tilde{A} + v\tilde{B})X$ were equal for $x = HX$, so ${H}^{t}(uA + vB)H = uA^{\prime} + vB^{\prime}$, then if $v_{1} = v(u,v)$ is a solution to ${v}^{t}(uA + vB) = 0$ with components homogeneous of degree m, it follows that $v_{2} = {H}^{-1}v_{1}$ is also homogeneous of degree m and $v_{2}(u\tilde{A} + v\tilde{B}) = 0$.

The NullSpace command in Mathematica provides a basis that is easily converted into Kronecker’s basis $v_{1},\ldots,v_{\mu }$.

The K-series is independent of its mode of construction. For example, as defined above using vectors (which Kronecker did not employ), the numbers m ₁, …, m _μ are independent of the choice of vectors $v_{1},v_{2},\ldots$ used to define them. See [240, v. 2, p. 38].

According to Jordan’s note of 2 March 1874 [325, p. 13].

Kronecker later called these transformations “Jacobi transformations,” as indicated in Section 5.1 and below in Section 5.6.4.

“zu dürftigen Inhalts” [358, p. 382].

The appendix arrived too late to be published in the 19 January proceedings and so appeared in the proceedings of 16 February 1874 [357]. See Section V [357, pp. 378–381]. A complete proof was of course not given, and it is unclear whether Kronecker had already worked out all the details. See in this connection Section 5.6.5 below.

For Kronecker’s criticisms, see [358, pp. 406–408]. At first, Jordan did not correctly understand the significance of the criticism and dismissed it, but in 1881, while working on his lectures at the Collège de France, he realized the import and validity of Kronecker’s criticism. In a note in the proceedings of the Paris Academy [329], he acknowledged his mistake and graciously attributed the first completely general reduction procedure to Kronecker.

Kronecker died in December 1891, without, apparently, ever having published his promised arithmetic theory. During December–January 1890–1891, he did publish a version of his algebraic theory as it applies to quadratic forms [368, 369], again for the purpose of comparison with his forthcoming arithmetic theory.

Although Kronecker’s reasoning implied the above-stated result, he focused instead on a table of integer invariants denoted by $(\mathfrak{J})$ [367, p. 151] derivable from the above W- and K-series and analogous to the table (J) he had introduced in his memoir on the special type of bilinear family u A + v A ^t discussed in Section 5.6.4.

Kronecker also generalized Weierstrass’ theory in another direction. He developed the entire theory outlined above for forms involving rx-variables and sy-variables, and so the matrices involved are r ×s. When the matrices are not square the lengths of the row and column K-series are generally different. Kronecker’s theory was elaborated by Muth in 1899 [450, pp. 93–133]. In the twentieth century, various approaches and refinements were introduced by Dickson [128], Turnbull [565], and Ledermann [405]. In his comprehensive treatise on the theory of matrices, Gantmacher devoted a chapter to Kronecker’s theory [240, V. 2, Ch. XII].

28.

R. Bölling, editor. Briefwechsel zwischen Karl Weierstrass und Sofja Kowalewskaja. Akademie Verlag, Berlin, 1993.MATH

30.

C. Borchardt. Neue Eigenschaft der Gleichung, mit deren Hülfe man die seculären Störungen der Planeten bestimmt. Jl. für die reine u. angew. Math., 30:38–45, 1846.MATHCrossRef

72.

A. L. Cauchy. Sur l’équation à l’aide de laquelle on determine les inégalités séculaires des mouvements des planètes. Exer. de math., 4, 1829. Reprinted in Oeuvres (2) 9, 174–195.

78.

A. L. Cauchy. Mémoire sur les perturbations produites dans les mouvements vibratoires d’un système de molécules par l’influence d’un autre système. Comptes rendus Acad. Sc. Paris, 30, 1850. Reprinted in Oeuvres (1) 4, 202–211.

90.

E. B. Christoffel. De motu permanenti electricitatis in corporibus homogenis. Dissertatio inauguralis. G. Shade, Berlin, 1856. Reprinted in Abhandlungen 1, 1–64.

91.

E. B. Christoffel. Verallgemeinerung einiger Theoreme des Herrn Weierstrass. Jl. für die reine u. angew. Math., 63:255–272, 1864. Reprinted in Abhandlungen 1, pp. 129–145.

92.

E. B. Christoffel. Über die kleinen Schwingungen eines periodisch eingerichteten Systems materieller Punkte. Jl. für die reine u. angew. Math., 63:273–288, 1864. Reprinted in Abhandlungen 1, 146–161.

93.

E. B. Christoffel. Theorie der bilinearen Functionen. Jl. für die reine u. angew. Math., 68: 253–272, 1868. Reprinted in Abhandlungen 1, 277–296.

94.

E. B. Christoffel. Über die Transformation der homogenen Differentialausdrücke zweiten Grades. Jl. für die reine u. angew. Math., 70:46–70, 1869. Reprinted in Abhandlungen 1, 352–377.

95.

A. Clebsch. Theorie der circularpolarisirenden Medien. Jl. für die reine u. angew. Math., 57:319–358, 1860.MATHCrossRef

98.

A. Clebsch. Über eine Classe von Gleichungen, welche nur reelle Wurzeln besitzen. Jl. für die reine u. angew. Math., 62:232–245, 1863.MATHCrossRef

126.

L. E. Dickson. History of the Theory of Numbers. Carnegie Institution of Washington, Washington, D.C., 1919–1923. 3 volumes.

128.

L. E. Dickson. Singular case of pairs of bilinear, quadratic, or Hermitian forms. Transactions of the American Mathematical Society, 29:239–253, 1927.MathSciNetMATHCrossRef

135.

P. G. Dirichlet. Recherches sur les formes quadratiques à coefficients et à indéterminées complexes. Jl. für die reine u. angew. Math., 24:291–371, 1842. Reprinted in Werke 1, 533–618.

202.

G. Frobenius. Gedächtnisrede auf Leopold Kronecker. Abhandlungen d. Akad. der Wiss. zu Berlin, pages 3–22, 1893. Reprinted in Abhandlungen 3, 707–724.

239.

E. Galois. Oeuvres mathématiques d’Évariste Galois. Jl. de math. pures et appl., 11:381–444, 1846. Reprinted as a book, Oeuvres mathématiques d’Évariste Galois, Paris, 1897.

240.

F. Gantmacher. Matrix Theory. AMS Chelsea Publishing, 2000. This work, published in two volumes, is an English translation of Gantmacher’s Teoriya Matrits (Moscow, 1953). It first appeared in 1959.

244.

C. Gauss. Disquisitiones arithmeticae. G. Fleischer, Leipzig, 1801. English translation by A. Clark (Yale University Press, New Haven, 1966). In quotations I have followed Clark’s translation unless otherwise noted.

246.

C. Gauss. Theoria residuorum biquadraticorum. Commentatio secunda. Comm. soc. sci. Gottingenesis, 7:93–149, 1832. Reprinted in Werke 2. Citations are to the German translation on pp. 534–586 of [248].

249.

C. F. Geiser. E. B. Christoffel. In E. B. Christoffel gesammelte mathematische Abhandlungen, volume 1, pages v–xv. B. G. Teubner, Leipzig, 1910.

286.

C. Hermite. Sur la décomposition d’un nombre en quatre carrés. Comptes Rendus, Acad. Sci. Paris, 37:133–134, 1853. Reprinted in Oeuvres 1, 288–289.

289.

C. Hermite. Sur la théorie des formes quadratiques. Seconde mémoire. Jl. für die reine u. angew. Math., 47:343–368, 1854. Reprinted in Oeuvres 1, 234–263.

290.

C. Hermite. Remarque sur un théorème de M. Cauchy. Comptes Rendus, Acad. Sci. Paris, 41:181–183, 1855. Reprinted in Oeuvres 1, 459–481.

310.

C. G. J. Jacobi. De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; … . Jl. für die reine u. angew. Math., 12:1–69, 1834. Reprinted in Werke 3, 191–268.

311.

C. G. J. Jacobi. De formatione et proprietatibus determinantium. Jl. für die reine u. angew. Math., 22:285–318, 1841. Reprinted in Werke 3, 255–392.

314.

C. G. J. Jacobi. Über eine elementare Transformation eines in Bezug auf jedes von zwei Variablen-Systemen linearen und homogenen Ausdrucks. Jl. für die reine u. angew. Math., 53:265–270, 1857. Reprinted in Werke 3, 583–590.

319.

F.-W. Janssen, editor. Elwin Bruno Christoffel. Gedenkschrift zur 150. Wiederkehr des Geburtstages. Kreis Aachen, 1979. Separate printing of Heitmatblätter des Kreises Aachen, Heft 3–4 (1978) and Heft 1 (1979).

320.

C. Jordan. Mémoire sur la résolution algébrique des équations. Journal de math. pures et appl., 12(2):109–157, 1867. Reprinted in Oeuvres 1, 109–157.

321.

C. Jordan. Sur la résolution algébrique des équations primitives de degré p ² (p étant premier impair). Journal de math. pures et appl., 13:111–135, 1868. Reprinted in Oeuvres 1, 171–202.

322.

C. Jordan. Traité des substitutions et des équations algébriques. Gauthier–Villars, Paris, 1870.

323.

C. Jordan. Sur la résolution des équations différentielles linéaires. Comptes Rendus, Acad. Sci. Paris, 73:787–791, 1871. Reprinted in Oeuvres 4, 313–317.

324.

C. Jordan. Sur les polynômes bilinéaires. Comptes Rendus, Acad. Sci. Paris, 77:1487–1491, 1873. Presented December 22, 1873. Reprinted in Oeuvres 3, 7–11.

325.

C. Jordan. Sur la réduction des formes bilinéaires. Comptes Rendus, Acad. Sci. Paris, 78: 614–617, 1874. Presented March 2, 1874. Reprinted in Oeuvres 3, 13–16.

326.

C. Jordan. Sur les systèmes de formes quadratiques. Comptes Rendus, Acad. Sci. Paris, 78:1763–1767, 1874. Reprinted in Oeuvres 3, 17–21.

327.

C. Jordan. Mémoire sur les formes bilinéaires. Jl. de math. pures et appl., 19:35–54, 1874. Submitted in August, 1873. Reprinted in Oeuvres 3, 23–42.

329.

C. Jordan. Observations sur la réduction simultanée de deux formes bilinéaires. Comptes Rendus, Acad. Sci. Paris, 92:1437–1438, 1881. Reprinted in Oeuvres 1, 189.

330.

C. Jordan. Cours d’analyse de l’École Polytechnique, volume 3. Gauthier–Villars, Paris, 1887.MATH

353.

L. Kronecker. Über bilineare Formen. Monatsberichte der Akademie der Wiss. zu Berlin, 1:145–162, 1866. Reprinted in Jl. für die reine u. angew. Math., 68:273–285 and in Werke 1, 145–162.

354.

L. Kronecker. Über Schaaren quadratischer Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 339–346, 1868. Reprinted in Werke 1, 163–174. The above title for the work was added by the editor (K. Hensel). See Werke 1, 163n. 1.

356.

L. Kronecker. Über Schaaren von quadratischen und bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 59–76, 1874. Presented Jan 19, 1874. Reprinted in Werke 1, 349–372.

357.

L. Kronecker. Über Schaaren von quadratischen und bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 149–156, 1874. Presented Feb 16, 1874. Reprinted in Werke 1, 373–381.

358.

L. Kronecker. Über Schaaren von quadratischen und bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 206–232, 1874. Presented March 16, 1874. Reprinted in Werke 1, 382–413.

359.

L. Kronecker. Über die congruenten Transformation der bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 397–447, 1874. Presented April 23, 1874. Reprinted in Werke 1, 423–483.

360.

L. Kronecker. Sur les faisceaux de formes quadratiques et bilinéaires. Comptes Rendus, Acad. Sci. Paris, 78:1181–1182, 1874. Presented April 27, 1874. Reprinted in Werke 1, 417–419.

367.

L. Kronecker. Algebraische Reduction der Schaaren bilinearer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1225–1237, 1890. Reprinted in Werke 3₂, 141–155.

368.

L. Kronecker. Algebraische Reduction der Schaaren quadratischer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1375–1388, 1890. Reprinted in Werke 3₂, 159–174.

369.

L. Kronecker. Algebraische Reduction der Schaaren quadratischer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 9–17, 34–44, 1891. Reprinted in Werke 3₂, 175–198.

405.

W. Ledermann. Reduction of singular pencils of matrices. Proceedings Edinburgh Mathematical Society, (2) 4:92–105, 1935.

441.

G. Mittag-Leffler. Weierstrass et Sonja Kowalewsky. Acta mathematica, 39:133–198, 1923.MathSciNetMATHCrossRef

450.

P. Muth. Theorie und Anwendung der Elementartheiler. Teubner, Leipzig, 1899.

565.

H. W. Turnbull. On the reduction of singular matrix pencils. Proceedings Edinburgh Mathematical Society, (2) 4:67–76, 1935.

587.

K. Weierstrass. Über ein die homogenen Functionen zweiten Grades betreffendes Theorem. Monatsberichte der Akademie der Wiss. zu Berlin, 1858. Reprinted in Werke 1, 233–246.

588.

K. Weierstrass. Zur Theorie des quadratischen und bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 311–338, 1868. Reprinted with modifications in Werke 2, 19–44.

591.

K. Weierstrass. Nachtrag zu der am 4. März … gelesenen Abhandlung: Über ein die homogenen Functionen zweiten Grades betreffendes Theorem. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 430–439, 1879. Reprinted with footnotes added by Weierstrass in Werke 3, 139–148.

616.

A. Yvon-Villarceau. Note sur les conditions des petites oscillations d’un corps solide de figure quelconque et la théorie des équations différentielles linéaires. Comptes rendus Acad. Sci. Paris, 71:762–766, 1870.

Titel: Further Development of the Paradigm: 1858–1874
verfasst von: Thomas Hawkins
Verlag: Springer New York
Buch: The Mathematics of Frobenius in Context
Print ISBN: 978-1-4614-6332-0

Electronic ISBN: 978-1-4614-6333-7

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-6333-7_5

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