Reference Request - Functions of the form $\frac{ax^2+bx+c}{dx^2+ex+f}$

Question

I'm looking for a refrence, an academic article, on functions of the form $\frac{ax^2+bx+c}{dx^2+ex+f}$
I know them as projective transformations of degree two, for example, because they describe the map of projecting from a conic to a line through a point not on the conic.

Answer 1

For lack of a standard term (that I’m aware of), I’ll sometimes call these functions “quadratic fractional transformations”, in analogy with “linear fractional transformations” for functions of the form $\frac{ax + b}{a'x + b'}$ that one typically encounters (with complex coefficients) in an introductory complex variables/analysis course, and which for real coefficients provide nice examples to illustrate precalculus graph transformation ideas. Most of the time, however, I’ll use the briefer term “quad/quad”.

A large number of school algebra books in the 1800s and very early 1900s include discussions of quad/quad functions, especially by French authors. I’ve mostly avoided listing such books, but I’ve included a few such books when the treatment is especially thorough and the book is probably not very well known. Thus, while I’m aware that you can find discussions of quad/quad functions in, for example, Hardy’s A Course of Pure Mathematics and in some or all of the algebra books by George Chrystal and Hall/Knight and Elias Loomis and Charles Smith and William Steadman Aldis and Isaac Todhunter, in the references below my focus is mostly on publications in periodical literature (i.e. journals).

For some reason the topic seems to be standard in older French examinations. It’s possible that Darboux’s 1869 paper below is partly responsible, but I suspect his paper arose from an existing fashion for this topic by teachers and exam question writers. One can find this topic in many past exams and exam syllabi in Nouvelles Annales de Mathématiques (1863 exam & 1871 syllabus & 1872 syllabus & 1873 syllabus $\ldots$ 1881 syllabus $\ldots$ 1893 exam $\ldots)$ and in Journal de Mathématiques Élémentaires (July 1878 exam & July 1879 exam & 1880 exam & 1883 exam $\ldots$ April 1892 exam $\ldots).$ About 15 years ago I went through the 1947−1962 volumes of Journal de Mathématiques Élémentaires (local university library has them), making extensive photocopies of things of interest to me, and most of these issues (24 issues per year) have detailed solutions to at least one set of examination questions in which a quad/quad problem appears, usually with specific numerical coefficients and often the problem includes an analytic-geometric component.

Quad/quad problems can also often be found in older competitive mathematics exams, such as the Cambridge Mathematical Tripos Examination and the Trinity Tripod Exam. For the former, in Wolstenholme’s 1878 book see bottom of p. 31 & #214, 215 on p. 33 & #217 on p. 34. For the latter, in Davison’s 1915 Subjects for Mathematical Essays see #103 on p. 113 (Google Books; Internet Archive). Incidentally, in Davison’s book see also pp. 24−25 & 93.

[1] Jean Gaston Darboux (1842−1917), Discussion de la fraction $\frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ [Discussion of the fraction $\frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$], Nouvelles Annales de Mathématiques (2) 8 (1869), pp. 81−86.

Google Books copy; Numdam copy

Google Translate of JFM 2.0279.01: The values of this expression are considered in the $3$ cases where $(2ac'+2ca'−bb')^2 - (b^2 - 4ac)(b'^2 - 4a'c') < 0,\;=0,\;>0,$ which leads to known geometric theorems.

[2] Jean Baptiste Étienne Vazeille (1825−1885), De l’involution [About involution], Journal de Mathématiques Élémentaires (1) 1 (1877), pp. 132−134 & 161−165.

Google Books copy pp. 132−134 & pp. 161−165; Internet Archive copy pp. 132−134 & pp. 161−165

Quadratic fractional transformations are discussed on pp. 162−165. The 2nd part ends on p. 165 with “(A suivre)” — to be continued — but I have not found a continuation.

[3] Auguste Morel (??−??), Note sur le trinomen et la fraction du second degré [Note on the trinomial and the fraction of the second degree], Journal de Mathématiques Élémentaires (1) 2 (1878), pp. 17−21.

Google Books copy; Internet Archive copy

[4] Pichenot (??−??), Remarques sur la fraction $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ [Notes on the fraction $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$], Journal de Mathématiques Élémentaires (1) 2 (1878), pp. 135−136.

Google Books copy; Internet Archive copy

Author identified at end of the volume, on p. 385 and on p. 393. I have not been able to determine with much confidence even a first-name initial.

[5] Maximilien François Célestin Henri Fajon (1826−??), Note d’algèbre [Algebra note], Journal de Mathématiques Élémentaires (1) 2 (1878), pp. 240−243.

Google Books copy; Internet Archive copy

[6] Maximilien François Célestin Henri Fajon (1826−??), Note d’algèbre [Algebra note], Journal de Mathématiques Élémentaires (1) 2 (1878), pp. 358−362.

Google Books copy; Internet Archive copy

[7] Charles Cochez (??−??), Recherche du maximum et du minimum de la fraction $z = \frac{ax^2 + 2bxy + cy^2}{a'x^2 + 2b'xy + c'y^2}$ [Research of the maximum and minimum of the fraction $z = \frac{ax^2 + 2bxy + cy^2}{a'x^2 + 2b'xy + c'y^2}$], Journal de Mathématiques Élémentaires (1) 3 (1879), pp. 230−232.

Google Books copy; Internet Archive copy

[8] Maximilien François Célestin Henri Fajon (1826−??), Variations des fonctions bicarrées déduites de celles des fonctions du second degré [Variations of bisquare functions deduced from those of functions of the second degree], Journal de Mathématiques Élémentaires et Spéciales (1) 4 (1880), pp. 205−210.

Google Books copy; Internet Archive copy

[9] Éd. Hénet (??−??), [Questions #1361], Nouvelles Annales de Mathématiques (2) 20 (1881), p. 144.

Google Books copy; Numdam copy

I don’t believe a solution to this problem was ever published, but it’s possible I overlooked such a solution. I do know that no solution was published by 1917 because the problem was reprinted on pp. 231−232 (Google Books; Numdam) of Anciennes questions non résolues [Old unresolved questions] in “Nouvelles Annales de Mathématiques” (4) 17 (1917), pp. 227−240.

[10] Auguste Morel (??−??), Note d’algèbre [Algebra note], Journal de Mathématiques Élémentaires et Spéciales (1) 5 (1881), pp. 388−392.

Google Books copy; Internet Archive copy

This paper is a follow-up of two publications (separately published pamphlets, I think) by Marcel Burat-Dubois (1833−??) — Note sur les questions de maximum et de minimum [Note on questions of maximum and minimum] (1881, 12 pages) and Règle pour determiner le maximum de la fraction $\frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ [Rule for determining the maximum of the fraction $\frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$] (1881, 17 pages), neither of which I’ve been able to locate digital copies (but see bottom right column here). The following papers below also appear to make nontrivial references to one or both of Burat-Dubois’s publications: Bourget (1881; pp. 438−442), Koehler (1881), Morel (1881; pp. 529−537), Lemoine (1881).

[11] Justin Bourget (1822−1887), Sur les variations de la fonction rationnelle du second degré $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ [On the variations of the rational function of the second degree $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$], Journal de Mathématiques Élémentaires et Spéciales (1) 5 (1881), pp. 438−442.

Google Books copy; Internet Archive copy

[12] Henri Camille Joseph Koehler (1837−1889), Correspondance, Journal de Mathématiques Élémentaires et Spéciales (1) 5 (1881), p. 480.

Google Books copy; Internet Archive copy

[13] Justin Bourget (1822−1887), Maxima et minima de la fonction rationnelle du second degré $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ [Maxima and minima of the rational function of the second degree $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$], Journal de Mathématiques Élémentaires et Spéciales (1) 5 (1881), pp. 481−486.

Google Books copy; Internet Archive copy

The footnote on p. 485 is by Auguste Morel (??−??).

[14] Auguste Morel (??−??), Étude élementaire sur les maxima et minima de la fraction du second degré [Elementary study on the maxima and minima of the fraction of the second degree], Journal de Mathématiques Élémentaires et Spéciales (1) 5 (1881), pp. 529−537.

Google Books copy; Internet Archive copy

[15] Émile Michel Hyacinthe Lemoine (1840−1912), Extrait d’une letter de M. Lemoine [Excerpt from a letter of M. Lemoine], Journal de Mathématiques Élémentaires et Spéciales (1) 5 (1881), pp. 548−550.

Google Books copy; Internet Archive copy

[16] Eugène Charles Catalan (1814−1894), Maximum et minimum de la fonction $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ [Maximum and minimum of the function $y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$], Mathesis Recueil Mathématique (1) 2 (1882), pp. 5−7.

Google Books copy

The footnote on p. 7 is by Paul Mansion (1844−1919).

[17] Émile Gelin (1850−1921), Questions de Mathématiques Élémentaires #9 [Elementary Mathematics Questions #9], Mathesis Recueil Mathématique (1) 3 (1883), p. 19.

Google Books copy

Gelin solves the following problem (translated from French): “Find a fraction of the form $$\frac{ax^2 + bx + c}{a'x^2 + b'x + c'}\,,$$ which has a maximum $m'$ corresponding to $x=x'$ and a minimum $m''$ corresponding to $x=x''.$”

[18] Justin Bourget (1822−1887), Maxima et minima du rapport de deux trinomes du second degré [Maxima and minima of the ratio of two quadratic trinomials], Journal de Mathématiques Élémentaires et Spéciales (2) 4 (1885), pp. 248−251.

Google Books copy; Internet Archive copy

[19] Joseph Jean Baptiste Neuberg (1840−1926), Notes Mathématiques #13 [Mathematics Notes #13], Mathesis Recueil Mathématique (1) 6 (1886), pp. 153−154.

Google Books copy

Neuberg gives some comments about Catalan’s 1882 paper above.

[20] Sohie/Denys/Decamps, [Solution to Question #672], Mathesis Recueil Mathématique (1) 10 (1890), pp. 149−150.

Google Books copy; Internet Archive copy

The problem solved is (translated from French): “Study the variation of the fraction $\frac{x^2 - 2ax + 3b}{x^2 + ax - 2b},$ $a$ and $b$ having arbitrary real values.”

[21] Charles Vacquant (1829−1895), Leçons d’Algèbre Élémentaire [Elementary Algebra Lessons], Librairie Charles Delagrave, 1894, ii + 655 pages.

Google Books copy; Hathitrust copy

Pages 445−467 gives a very lengthy treatment of quadratic fractional transformations.

[22] Charles [Carlo] Émile Ernest Bourlet (1866−1913), Leçons d’Algèbre Élémentaire, Armand Colin & Cie (Mézières, France), 1896, xii + 518 pages.

Google Books copy; Internet Archive copy

See Article 180 — Variation de la fraction du second degré [Variation of the fraction of the second degree] — on pp. 529−541, and the first footnote on p. vii (Google Books; Internet Archive).

[23] Alfred Lodge (1854−1937) and William E. Jeffares (1833?−1898), [Solution to Problem #80], Mathematical Gazette 1 #24 (December 1900), pp. 417−418.

Google Books copy; Internet Archive copy; JSTOR copy

Statement of Problem #80, incorporating corrections given by Lodge: “Prove that $\frac{x+a}{x^2 + bx + c}$ will always lie between two fixed finite limits if $b^2 < 4c$; that there will be two limits between which it cannot lie if $a^2 + c > ab$ and $b^2 > 4c$; and that it will be capable of all values if $a^2 + c < ab.$” Note: This is linear/quadratic, but since everything thus far has been in French, I thought it worth while to include this item despite my having omitted many other items in French that deal with linear/quadratic or quadratic/linear.

[24] Pierre René Jean Baptiste Henri Brocard (1845−1922), Correspondence [23 November 1900], L'Enseignement Mathématique 3 (1901), pp. 59−60.

Google Books copy; SwissDML copy

[25] Paul Jean Joseph Barbarin (1855−1931), Sur une variation élémentaire [On an elementary variation], L'Enseignement Mathématique 3 (1901), pp. 216−218.

Google Books copy; SwissDML copy

[26] George Greenhill (1847−1927), Sur une variation élémentaire [On an elementary variation], L'Enseignement Mathématique 3 (1901), pp. 328−333.

Google Books copy; SwissDML copy

Google Translate of JFM 32.0176.06: From p. 59 of the same volume of the Enseignement mathématique 1901, H. Brocard had taken a position against the custom common in French lyceums, that of quadratic substitution $$y = \frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$$ defined relationship even before the theory of conic sections and differential calculus become the subject of investigation. In reply to this communication, Barbarin (see JFM 32.0176.05) draws attention to an elementary method of Hermite's which allows the investigation of his dependency without resorting to the concept of derivation. Greenhill shares another, even simpler, method. At the same time, he points out the usefulness that this substitution offers when integrating expressions that contain the quadratic irrationality $\sqrt{ax^2 + bx + c}\,.$

[27] George Greenhill (1847−1927) and Eugène Charles Catalan (1814−1894), Sur la fraction $\frac{ax^2 + bx + c}{Ax^2 + Bx + C}$ [On the fraction $\frac{ax^2 + bx + c}{Ax^2 + Bx + C}$], Mathesis Recueil Mathématique (3) 1 (1901), pp. 268−270.

Google Books copy

This is an excerpt from Greenhill’s “L'Enseignement Mathématique” paper (pp. 268−269) and some comments by Catalan (p. 270).

[28] Boleslas Alexandre Niewenglowski (1846−1933), Cours D’Algèbre [Course of Algebra], Volume 2, 5th edition, Librairie Armand Colin (Paris), 1902, iii + 488 pages.

Google Books copy

See Section 14 — Variations de $\frac{ax^2 + bx + c}{a'x^2 + b'x + c'}$ — on pp. 81−82.

[29] Ch. B (??−??), Sur la courbe qui représente les variations du rapport de deux trinomes du second degré [On the curve which represents the variations of the ratio of two quadratic trinomials], Journal de Mathématiques Élémentaires 31 #10 (15 February 1907), p. 73.

Google Books copy

[30] Joseph Jean Baptiste Neuberg (1840−1926), Sur les maxima et les minima [On maxima and minima], Notes Mathématiques #6, Mathesis Recueil Mathématique (3) 7 (1907), p. 70.

Google Books copy

[31] Joseph Jean Baptiste Neuberg (1840−1926), Review of “Théorie et Applications des Équations du Second Degré” by Juhel-Rénoy, Mathesis Recueil Mathématique (3) 9 (1909), pp. 69−70.

Google Books copy

Neuberg’s review suggests that this book is worth looking at, as well as this Amer. Math. Monthly review. However, I have not found a digital copy of the book. Also, a 1920 2nd edition was published — see Bull. AMS announcemnt and JFM announcement.

[32] Jean Baptiste Pomey (1861−1943), Sur une propriété de la fraction rationnelle du second degré [On a property of the rational fraction of the second degree], Nouvelles Annales de Mathématiques (4) 17 (1917), pp. 441−448.

Google Books copy; Numdam copy

[33] Joseph Leslie Riley (1880−1960) and Aloysius Francis Frumveller (1872−1950), Problem 2793, American Mathematical Monthly 28 #3 (March 1921), p. 146.

Google Books copy; JSTOR copy

Proposed by Riley, discussion by Frumveller: “If $a,$ $b,$ and $c,$ are complex, and $\alpha,$ $\beta,$ and $\gamma,$ real constants, the point $$x = \frac{at^2 + 2bt + c}{\alpha t^2 + 2\beta t + \gamma}$$ traces a conic or a straight line when $t$ takes all real values.”

[34] Hugh Vernon Lowry (1897−??), A rule and compass method of finding the maximum and minimum values of $(ax^2 + 2bx + c)/(a'x^2 + 2b'x + c'),$ Mathematical Gazette 12 #170 (May 1924), pp. 110−111.

Google Books copy; JSTOR copy

[35] Clement Vavasor Durell (1882−1968), Advanced Algebra, Volume I, George Bell and Sons, 1932, viii + 193 + i-xxii pages.

Internet Archive copy

See The Function $\frac{ax^2 + 2bx + c}{Ax^2 + 2Bx + C}$ on pp. 145−152.

[36] Norman Martin Gibbins (1882−1956), The quadratic quotient, Mathematical Gazette 20 #237 (February 1936), pp. 53−55.

JSTOR copy

[37] Robert Franklin Muirhead (1860−1941), An algebraic note, Mathematical Gazette 23 #257 (December 1939), pp. 471−473.

JSTOR copy

[38] R. Holmes (??−??), On note 1429, Mathematical Gazette 24 #259 (May 1940), pp. 123−124.

JSTOR copy.

“Note 1429” is Muirhead’s 1939 paper above.

[39] George William Brewster (1881−??), On Note 1429 (Gazette, XXIII, p. 471), Mathematical Gazette 24 #261 (October 1940), pp. 290−291.

JSTOR copy

[40] A. A. Krishnaswami Ayyangar (1892−1953), On notes 1429 and 1457, Mathematical Gazette 33 #304 (May 1949), pp. 123−125.

JSTOR copy

“Note 1429” and “Note 1457” are Muirhead’s 1939 paper and Holmes’s 1940 paper.

[41] Henry Thomas Herbert Piaggio (1884−1967), The quotient of two quadratic functions, Mathematical Gazette 36 #317 (September 1952), 208−209.

JSTOR copy

[42] R. D. Lord (??−??), The ratio of two quadratics, Mathematical Gazette 37 #322 (December 1953), pp. 271−273.

JSTOR copy

This paper is a follow-up to Ayyangar’s 1949 paper above.

[43] C. V. Gregg (??−??), The quotient of two quadratic functions, Mathematical Gazette 39 #327 (February 1955), pp. 50−52.

JSTOR copy

This paper is a follow-up to Piaggio’s 1952 paper above.

[44] C. V. Gregg (??−??), The quotient of two quadratics, Mathematical Gazette 39 #330 (December 1955), pp. 312−313.

JSTOR copy

[45] Stefan Straszewicz (1889−1983), Mathematical Problems And Puzzles from the Polish Mathematical Olympiads, translated by Janina Smólska (1917−2002), Pergamon Press, 1965, viii + 367 pages.

Internet Archive copy

Pages 34−35 gives a non-calculus method of solving problem #25 (on p. 5) — “Find the least value of the fraction $\frac{x^4 + x^2 + 5}{(x^2 + 1)^2}\,$”, after which follows a lengthy REMARK on pp. 35−37 that discusses the following — “We shall solve a more general problem: find the least and the greatest values of the function $$y = \frac{x^2 + mx + n}{x^2 + px + q}$$ under the assumption that the trinomials $x^2 + mx + n$ and $x^2 + px + q$ have no root in common.” Note: Straszewicz is Smólska’s father.