Celebratio Mathematica

G. Bliss: “The motion of a heavenly body in a resisting medium,” Pop. Astron. 6 (1898), pp. 20–29.

G. A. Bliss: “The second variation of a definite integral when one end-point is variable,” Trans. Amer. Math. Soc. 3 : 1 (1902), pp. 132–141. MR 1500591 JFM 33.0385.01

G. A. Bliss: “The geodesic lines on the anchor ring,” Ann. of Math. (2) 4 : 1 (October 1902), pp. 1–21. MR 1502291 JFM 33.0670.02

G. A. Bliss: “Jacobi’s criterion when both end-points are variable,” Math. Ann. 58 : 1–2 (1903), pp. 70–80. MR 1511229 JFM 34.0402.01

G. A. Bliss: “An existence theorem for a differential equation of the second order, with an application to the Calculus of Variations,” Trans. Amer. Math. Soc. 5 : 2 (1904), pp. 113–125. MR 1500665 JFM 35.0340.01

G. A. Bliss: “Sufficient condition for a minimum with respect to one-sided variations,” Trans. Amer. Math. Soc. 5 : 4 (1904), pp. 477–492. MR 1500686 JFM 35.0370.02

In a problem of the Calculus of Variations with fixed endpoints, the curves along which the integral is taken may be free to traverse any portion of the plane, or they may be restricted to lie in a given region \( R \). In the latter case it sometimes happens that although no solution can be found which lies entirely within the region, yet one exists consisting partly of interior arcs and partly of arcs of the boundary. In the neighborhood of the boundary arcs the only variations considered must be of course interior to the region, and therefore lie on one side only of the curve whose minimizing properties are to be investigated.

The necessary conditions which such a combination of interior curve and boundary must satisfy in order to minimize the given integral have already been derived. It is proposed in the following paper to prove that these conditions with slight modification are also sufficient: first, when the integral has the form \[ I = \int f(x,y,y^{\prime})\,dx; \] and second, when the integral is \[ I = \int f(x,y,x^{\prime},y^{\prime})\,dt, \] taken along a curve \[ x = \mu(t),\quad y = \nu(t) \] in parameter representation.

G. A. Bliss: “The exterior and interior of a plane curve,” Bull. Amer. Math. Soc. 10 : 8 (1904), pp. 398–404. MR 1558133 JFM 35.0506.01

The theorem that a continuous closed curve without double points divides the plane into two regions, an exterior and an interior, has been discussed by several writers. Jordan [1882, p. 91–99] and Schoenflies [1896, p. 85] assume the theorem for polygons and then extend it to apply to more general curves by processes which are extremely complicated. In the March number of the Bulletin, Ames has sketched an apparently simpler method which applies to the so-called regular curves. The writer ventures to give below another proof for a class of curves which is more general in that the tangent is not assumed to exist, but which on the other hand may not include all of the regular curves. Polygons, curves consisting of a finite number of analytic pieces, and regular curves of the type treated by Schoenflies, are among those to which the method applies.

G. A. Bliss: “The solutions of differential equations of the first order as functions of their initial values,” Ann. of Math. (2) 6 : 2 (January 1905), pp. 49–68. MR 1503546 JFM 36.0389.01

The proof of the existence of solutions of the differential equation \[ \frac{dx}{dt} = f(t,x), \] where the funtion \( f \) is expansible into a power-series in \( t \) and \( x \), is included in most of the text books on differential equations. Methods have also been devised for proving the existence of a integral taking a given value \( \xi \) for \( t=\tau \), when the function \( f \) satsifies much less stringent restrictions [Osgood 1898]. In many applications however the mere existence theorem is of little value unless accompanied by a proof of the continuity and differentiability of the solution considered as a function of the initial constants \( \tau, \xi \). As the literature dealing with these latter questions is somewhat scattered and the proofs complicated, it is proposed in the following pages to collect and simplify as far as possible the results of the different writers. Solutions are defined reaching from boundary to boundary of the region \( R \) over which the properties of the function \( f \) are assumed provided \( R \) is closed, and the method of Peano for proving the existence and continuity of the first partial derivatives with respect to the initial constants has been extended by a simple device to the higher derivatives.

G. A. Bliss: “A generalization of the notion of angle,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 184–196. MR 1500741 JFM 37.0490.01

G. A. Bliss and M. Mason: “A problem of the Calculus of Variations in which the integrand is discontinuous,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 325–336. MR 1500752 JFM 37.0402.01

G. A. Bliss: “A proof of the fundamental theorem of analysis situs,” Bull. Amer. Math. Soc. 12 : 7 (1906), pp. 336–341. MR 1558345 JFM 37.0495.02

The theorem that a Jordan curve divides the plane into two regions, an interior and an exterior, has in recent years received much attention. The proofs which have been given may be roughly divided into two classes, those in which the object has been to prove the theorem with the fewest possible hypotheses on the curve [Veblen 1905], and those in which generality has to a certain extent been sacrificed for simplicity [Ames 1905, p. 353; Bliss 1904, p. 398]. The following proof belongs to the second class. In §1 it is assumed that the curve considered is continuous and has a continuously turning tangent at every point; but in §3, by extending the proof of one of the auxiliary theorems, curves with a finite number of corner points and singularities are also included. The proof is presented here because it seems shorter and simpler than those heretofore given.

G. A. Bliss: “Book review: The theory of functions of real variables, by James Pierpont,” Bull. Amer. Math. Soc. 13 : 3 (1906), pp. 119–130. MR 1558414

G. A. Bliss: “A new form of the simplest problem of the Calculus of Variations,” Trans. Amer. Math. Soc. 8 : 3 (1907), pp. 405–414. MR 1500795 JFM 38.0408.01

G. A. Bliss: “The construction of a field of extremals about a given point,” Bull. Amer. Math. Soc. 13 : 7 (1907), pp. 321–324. MR 1558473 JFM 38.0408.02

M. Mason and G. A. Bliss: “The properties of curves in space which minimize a definite integral,” Trans. Amer. Math. Soc. 9 : 4 (1908), pp. 440–466. MR 1500821 JFM 39.0441.01

G. A. Bliss: “On the inverse problem of the Calculus of Variations,” Ann. of Math. (2) 9 : 3 (April 1908), pp. 127–140. MR 1502361 JFM 39.0445.01

G. A. Bliss: “A method of deriving Euler’s equation in the Calculus of Variations,” Amer. Math. Monthly 15 : 3 (March 1908), pp. 47–54. MR 1516998 JFM 40.1031.08

The derivation of Euler’s equation can be made by means of integrals of the form \begin{equation*}\tag{1} \int[A(x,y) + B(x,y)y^{\prime}]\,dx \end{equation*} (with \( y^{\prime} = dy/dx \)), which are independent of the path of integration. Invariant integrals of this type play an important role in the proofs that the usual conditions in the Calculus of Variations are sufficient to insure a minimum or a maximum, and their introduction in connection with the derivation of Euler’s equation makes it possible to simplify considerably the presentation of the whole theory.

In the first section below there is a simple discussion of the conditions under which an integral of the form (1) is independent of the path, and in the second section these results are applied to the derivation of Euler’s equation.

G. A. Bliss and M. Mason: “Fields of extremals in space,” Trans. Amer. Math. Soc. 11 : 3 (1910), pp. 325–340. MR 1500866 JFM 41.0437.01

G. A. Bliss: “Book review: An introduction to the study of integral equations, by Maxime Bôcher,” Bull. Amer. Math. Soc. 16 : 4 (1910), pp. 207–213. MR 1558889

G. A. Bliss: “A new proof of Weierstrass’s theorem concerning the factorization of a power series,” Bull. Amer. Math. Soc. 16 : 7 (1910), pp. 356–359. MR 1558920 JFM 41.0286.03

G. A. Bliss: “The function concept and the fundamental notions of the Calculus,” pp. 263–304 in Monographs on topics of modern mathematics relevant to the elementary field. Edited by J. W. A. Young. Longmans Green and Co., 1911. Zbl 0067.03101

G. A. Bliss: “A generalization of Weierstrass’ preparation theorem for a power series in several variables,” Trans. Amer. Math. Soc. 13 : 2 (April 1912), pp. 133–145. MR 1500910 JFM 43.0504.02

G. A. Bliss: “A new proof of the existence theorem for implicit functions,” Bull. Amer. Math. Soc. 18 : 4 (1912), pp. 175–179. MR 1559180 JFM 43.0483.04

G. A. Bliss: “Fundamental existence theorems,” pp. i–ii,1–107 in The Princeton Colloquium (Princeton University, 15–17 September 1909), part 1. American Mathematical Society (New York), 1913. JFM 44.0365.01

G. A. Bliss and F. B. Wiley: “A method of subdividing the interior of a simply closed rectifiable curve with an application to Cauchy’s theorem,” Bull. Sci. Lab. Denison Univ. 17 (1914), pp. 375–389.

G. A. Bliss and A. L. Underhill: “The minimum of a definite integral for unilateral variations in space,” Trans. Amer. Math. Soc. 15 : 3 (1914), pp. 291–310. MR 1500981 JFM 45.0605.03

G. A. Bliss: “The Weierstrass \( E \)-function for problems of the Calculus of Variations in space,” Trans. Amer. Math. Soc. 15 : 4 (October 1914), pp. 369–378. MR 1500985 JFM 45.0604.03

G. A. Bliss: “A note on symmetric matrices,” Ann. of Math. (2) 16 : 1–4 (1914–1915), pp. 43–44. MR 1502486 JFM 45.0262.05

G. A. Bliss: “A substitute for Duhamel’s theorem,” Ann. of Math. (2) 16 : 1–4 (1914–1915), pp. 45–49. MR 1502487 JFM 45.0452.02

G. A. Bliss: “A note on functions of lines,” Proc. Natl. Acad. Sci. USA 1 : 3 (March 1915), pp. 173–177. JFM 45.0550.01

G. A. Bliss: “Generalizations of geodesic curvature and a theorem of Gauss concerning geodesic triangles,” Amer. J. Math. 37 : 1 (January 1915), pp. 1–18. MR 1507894 JFM 45.0859.02

G. A. Bliss: “Book review: Leçons sur les fonctions des lignes, by Vito Volterra,” Bull. Amer. Math. Soc. 21 : 7 (1915), pp. 345–355. MR 1559651

G. A. Bliss: “Jacobi’s condition for problems of the Calculus of Variations in parametric form,” Trans. Amer. Math. Soc. 17 : 2 (April 1916), pp. 195–206. MR 1501037 JFM 46.0758.03

G. A. Bliss: “A note on the problem of Lagrange in the Calculus of Variations,” Bull. Amer. Math. Soc. 22 : 5 (1916), pp. 220–225. MR 1559765 JFM 45.0605.01

There are two theorems concerning the solutions of a system of linear differential equations, due to von Escherich and A. Mayer, which play an important rôle in the proofs of the sufficient conditions for a problem of the Calculus of Variations in the form proposed by Lagrange. Bolza remarks [1933, p. 634] that the theory of the second variation has so far been essential to the establishment of these two theorems, as well as to the proof of the necessity of Jacobi’s condition in the exceptional cases not covered by the geometrical theory of Kneser. In a paper which will appear in the near future, the writer has shown that for problems in parametric form in any number of dimensions an inclusive proof of the Jacobi condition may be made very simply by an application of Euler’s equations and the usual corner point condition to the second variation. A similar result has been attained for the problem of Lagrange by D. M. Smith. It is desirable therefore to have deductions of the two theorems mentioned above which also shall be independent of the complicated transformations of the second variation. The proofs given below have this advantage.

G. A. Bliss: “Book review: Topics in the theory of functions of several complex variables, by William Fogg Osgood,” Bull. Amer. Math. Soc. 23 : 1 (1916), pp. 35–44. MR 1559859

G. A. Bliss: “A necessary and sufficient condition for the existence of a Stieltjes integral,” Proc. Natl. Acad. Sci. USA 3 : 11 (November 1917), pp. 633–637. JFM 46.1458.03

G. A. Bliss: “Integrals of Lebesgue,” Bull. Amer. Math. Soc. 24 : 1 (1917), pp. 1–47. MR 1560001 JFM 46.0383.01

G. A. Bliss: “The problem of Mayer with variable end points,” Trans. Amer. Math. Soc. 19 : 3 (1918), pp. 305–314. MR 1501104 JFM 46.0758.04

G. A. Bliss: “Solutions of differential equations as functions of the constants of integration,” Bull. Amer. Math. Soc. 25 : 1 (1918), pp. 15–26. MR 1560139 JFM 47.0940.01

The purpose of this note is to prove the differentiability of the solutions of a system of differential equations with respect to the constants of integration by a method which seems more natural and simpler than those which have hitherto been published. Incidentally a restatement of the so-called “imbedding theorem” for differential equations is given, a theorem which is frequently applied in the Calculus of Variations, and which has been useful, and could be made still more so, in many other connections. It is analogous to the fundamental theorem for implicit functions in its statement that a solution of a system of differential equations given in advance is always a member of a continuous family of such solutions.

G. A. Bliss: “A method of computing differential corrections for a trajectory,” J. U.S. Artillery 51 (1919), pp. 445–449. Corrected copy of earlier printing in vol. 50 (pp. 455–460).

G. A. Bliss: “The use of adjoint systems in the problem of differential corrections for a trajectory,” J. U.S. Artillery 51 (1919), pp. 296–311.

G. A. Bliss: Differential corrections for anti-aircraft guns. Technical report, Aberdeen Proving Ground, 1919.

G. A. Bliss: “Differential equations containing arbitrary functions,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 79–92. MR 1501137

G. A. Bliss: “Functions of lines in ballistics,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 93–106. MR 1501138 JFM 47.0382.02

G. A. Bliss: “Some recent developments in the Calculus of Variations,” Bull. Amer. Math. Soc. 26 : 8 (1920), pp. 343–361. MR 1560316 JFM 47.0471.04

G. A. Bliss: “Birational transformations simplifying singularities of algebraic curves,” Trans. Amer. Math. Soc. 24 : 4 (1922), pp. 274–285. MR 1501226 JFM 50.0266.02

G. A. Bliss: “The reduction of singularities of plane curves by birational transformation,” Bull. Amer. Math. Soc. 29 : 4 (1923), pp. 161–183. Presidential address delivered before the American Mathematical Society, December 28, 1922. MR 1560693 JFM 49.0264.01

G. A. Bliss: “Book review: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein’s Relativitätstheorie, vol. 1: Elementare Differentialgeometrie, by W. Blaschke,” Bull. Amer. Math. Soc. 29 : 7 (1923), pp. 322–325. MR 1560739

G. A. Bliss: “Algebraic functions and their divisors,” Ann. of Math. (2) 26 : 1–2 (September–October 1924), pp. 95–124. MR 1502680 JFM 50.0699.03

G. A. Bliss: “A boundary value problem for a system of ordinary linear differential equations of the first order,” Trans. Amer. Math. Soc. 28 : 4 (1926), pp. 561–584. MR 1501366 JFM 52.0453.13

G. A. Bliss: “A boundary value problem in the Calculus of Variations,” Bull. Amer. Math. Soc. 32 : 4 (1926), pp. 317–331. MR 1561219 JFM 52.0509.02

G. A. Bliss: “Book review: Elementi del calcolo delle variazioni, by G. Vivanti,” Bull. Amer. Math. Soc. 32 : 4 (1926), pp. 392–393. MR 1561233

G. A. Bliss: “Contributions that have been made by pure science to the advancement of engineering and industry. Mathematics.,” Sci. Mon. 24 (1927), pp. 308–319. JFM 52.0036.11

G. Bliss: “The transformation of Clebsch in the Calculus of Variations,” pp. 589–603 in Proceedings of the International Mathematical Congress (Toronto, 11–16 August 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press, 1928. JFM 54.0532.03

G. A. Bliss: “Book review: Calculus of Variations, by A. R. Forsyth,” Bull. Amer. Math. Soc. 34 : 4 (1928), pp. 512–514. MR 1561597

G. A. Bliss: “The problem of Lagrange in the Calculus of Variations,” Amer. J. Math. 52 : 4 (October 1930), pp. 673–744. MR 1506783 JFM 56.0435.01

G. A. Bliss: “An integral inequality,” J. London Math. Soc. S1-5 : 1 (1930), pp. 40–46. MR 1574997 JFM 56.0434.02

G. A. Bliss and I. J. Schoenberg: “On separation, comparison and oscillation theorems for self-adjoint systems of linear second order differential equations,” Amer. J. Math. 53 : 4 (October 1931), pp. 781–800. MR 1506854 JFM 57.0528.01 Zbl 0003.25702

Contributions to the Calculus of Variations, 1930. Edited by G. A. Bliss and L. M. Graves. University of Chicago Press, 1931. Zbl 0003.40002

Topics of the Calculus of Variations. Edited by G. Bliss. University of Chicago, 1932. Principally a report on lectures by E. J. McShane and Richard Courant.

Edward James McShane	Related
Richard Courant	Related

G. A. Bliss: “Ernest Julius Wilczynski,” Science 76 : 1971 (October 1932), pp. 316–317. JFM 58.0995.07

G. A. Bliss: “The problem of Bolza in the Calculus of Variations,” Ann. of Math. (2) 33 : 2 (April 1932), pp. 261–274. MR 1503050 JFM 58.0535.02 Zbl 0004.15502

G. A. Bliss: “The Calculus of Variations and the quantum theory,” Bull. Amer. Math. Soc. 38 : 4 (1932), pp. 201–224. MR 1562365 JFM 58.0530.01 Zbl 0004.15601

G. A. Bliss and I. J. Schoenberg: “On the derivation of necessary conditions for the problem of Bolza,” Bull. Amer. Math. Soc. 38 : 12 (1932), pp. 858–864. MR 1562533 JFM 58.0535.01 Zbl 0006.25902

G. Bliss: Variationsrechnung. Edited by F. Schwank. B. G. Teubner (Leipzig and Berlin), 1932. JFM 58.0529.08 Zbl 0003.34803

G. A. Bliss: “Eliakim Hastings Moore,” Univ. Rec. 19 (1933), pp. 130–134.

G. A. Bliss and M. R. Hestenes: “Sufficient conditions for a problem of Mayer in the Calculus of Variations,” Trans. Amer. Math. Soc. 35 : 1 (1933), pp. 305–326. MR 1501685 JFM 59.0497.02 Zbl 0006.25903

G. A. Bliss: “Mathematical interpretations of geometrical and physical phenomena,” Amer. Math. Monthly 40 : 8 (October 1933), pp. 472–480. Reprinted as Chapter 8 of A mathematician explains by M. I. Logsdon (University of Chicago Press, 1936) pp. 146–158. MR 1522896 JFM 59.0060.01

G. A. Bliss: “Eliakim Hastings Moore,” Bull. Amer. Math. Soc. 39 : 11 (1933), pp. 831–838. MR 1562740 JFM 59.0038.02

G. A. Bliss: The Calculus of Variations, multiple integrals. University of Chicago, 1933. MR 2936825

Contributions to the Calculus of Variations, 1931–32. Edited by G. A. Bliss and L. M. Graves. University of Chicago Press, 1933. Zbl 0006.40401

G. A. Bliss: Algebraic functions. AMS Colloquium Publications 16. American Mathematical Society (New York), 1933. JFM 59.0384.03 Zbl 0008.21004

G. A. Bliss: The Calculus of Variations in three space. University of Chicago, 1934. Lecture notes, Spring 1934.

G. A. Bliss: “The scientific work of Eliakim Hastings Moore,” Bull. Amer. Math. Soc. 40 : 7 (1934), pp. 501–514. MR 1562892

G. A. Bliss: The problem of Bolza in the Calculus of Variations. University of Chicago, 1935. Lecture notes, Winter 1935.

G. A. Bliss: “Book review: ‘Inequalities’, by Hardy, Littlewood, and Pólya,” Science 81 (1935), pp. 565–566.

J. E. Littlewood	Related
Godfrey Harold Hardy	Related
George Pólya	Related

G. A. Bliss and L. E. Dickson: “Biographical memoir of Eliakim Hastings Moore, 1862–1932,” Proc. Natl. Acad. Sci. USA 17 (1936), pp. 83–102.

L. E. Dickson	Related	Volume
E. H. Moore	Related	Volume

G. A. Bliss: “The evolution of problems of the Calculus of Variations,” Amer. Math. Monthly 43 : 10 (December 1936), pp. 598–609. MR 1523791 Zbl 0015.35702

Contributions to the Calculus of Variations, 1933–37. Edited by G. A. Bliss, L. M. Graves, and W. T. Reid. University of Chicago Press, 1937.

Lawrence Murray Graves	Related
William Thomas Reid	Related

G. A. Bliss and L. E. Dickson: “Herbert Ellsworth Slaught,” Science 86 (1937), pp. 72–73.

L. E. Dickson	Related	Volume
Herbert Ellsworth Slaught	Related

G. A. Bliss: “Herbert Ellsworth Slaught — In memoriam,” Bull. Amer. Math. Soc. 43 : 9 (1937), pp. 595–597. MR 1563596

G. A. Bliss: “Normality and abnormality in the Calculus of Variations,” Trans. Amer. Math. Soc. 43 : 3 (1938), pp. 365–376. MR 1501950 JFM 64.0510.01 Zbl 0019.12303

G. A. Bliss: “Definitely self-adjoint boundary value problems,” Trans. Amer. Math. Soc. 44 : 3 (1938), pp. 413–428. MR 1501974 JFM 64.0438.02 Zbl 0020.03204

G. A. Bliss: “Herbert Ellsworth Slaught — teacher and friend,” Amer. Math. Monthly 45 : 1 (January 1938), pp. 5–10. MR 1524156 JFM 64.0023.14

Contributions to the Calculus of Variations, 1938–41. Edited by G. A. Bliss, M. R. Hestenes, and W. T. Reid. University of Chicago Press, 1942.

William Thomas Reid	Related
Magnus Rudolph Hestenes	Related

G. A. Bliss: “The Calculus of Variations for multiple integrals,” Amer. Math. Monthly 49 : 2 (February 1942), pp. 77–89. MR 0006022 Zbl 0063.00455

G. A. Bliss: “Oskar Bolza,” Science 97 : 2509 (1943), pp. 108–109. article

G. A. Bliss: Mathematics for exterior ballistics. John Wiley and Sons (New York), 1944. MR 0010480 Zbl 0063.00457

G. A. Bliss: “Oskar Bolza — in memoriam,” Bull. Amer. Math. Soc. 50 : 7 (1944), pp. 478–489. MR 1564626

G. A. Bliss: Lectures on the calculus of variations. University of Chicago Press (Chicago, IL), 1946. MR 0017881 Zbl 0063.00459

G. A. Bliss: Calculus of Variations, 6th edition. Carus Mathematical Monographs 1. The Mathematical Association of America (Washington, DC), 1971. Zbl 0317.49001

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Gilbert Ames Bliss

Gilbert Ames Bliss
May 9, 1876 – May 8, 1951
A Biographical Memoir

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Gilbert Ames BlissMay 9, 1876 – May 8, 1951A Biographical Memoir

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Gilbert Ames Bliss
May 9, 1876 – May 8, 1951
A Biographical Memoir