1Euler, the first systematic writer on the ‘integral calculus’, defined it in a manner which identifies it with the theory of differential equations: ‘calculus integralis est methodus, ex data differentialium relatione inveniendi relationem ipsarum quantitatum’ (Institutiones calculi integralis, p. 1). We are concerned only with the special equation , but all the remarks we have made may be generalised so as to apply to the wider theory.
2An algebraical number is a number which is the root of an algebraical equation whose coefficients are integral. It is known that there are numbers (such as and ) which are not roots of any such equation. See, for example, Hobson’s Squaring the circle (Cambridge, 1913).
3For fuller information the reader may be referred to Appell and Goursat’s Théorie des fonctions algébriques.
4‘Mémoire sur la classification des transcendantes, et sur l’impossibilité d’exprimer les racines de certaines équations en fonction finie explicite des coefficients’, Journal de mathématiques, ser. 1, vol. 2, 1837, pp. 56–104; ‘Suite du mémoire…’, ibid. vol. 3, 1838, pp. 523–546.
5The natural generalisations of the theory of algebraical equations are to be found in parts of the theory of differential equations. See Königsberger, ‘Bemerkungen zu Liouville’s Classificirung der Transcendenten’, Math. Annalen, vol. 28, 1886, pp. 483–492.
6For example, cannot be equal to , where is an algebraical function of .
7Théorie analytique des probabilités, p. 7.
8See, e.g., Weber’s Traité d’algèbre supérieure (French translation by J. Griess, Paris, 1898), vol. 1, pp. 61–64, 143–149, 350–353; or Chrystal’s Algebra, vol. 1, pp. 151–162.
9The proof will be completed in v., 16.
10The following account of Hermite’s method is taken in substance from Goursat’s Cours d’analyse mathématique (first edition), t. 1, pp. 238–241.
11See Chrystal’s Algebra, vol. 1, pp. 119 et seq.
12See, for example, Hardy, A course of pure mathematics (2nd edition), p. 208.
13The operation of forming the derived function of a given polynomial can of course be effected by a combination of these operations.
14It is easy to show that the solution is also unique.
15See Cajori, An introduction to the modern theory of equations (Macmillan, 1904); Mathews, Algebraic equations (Cambridge tracts in mathematics, no. 6), pp. 6–7.
16The equation is soluble by radicals in certain cases. See Mathews, l.c., pp. 52 et seq.
17We now write for for the sake of symmetry in notation.
18See, for example, Bromwich, l.c., pp. 16 et seq.
19Cf. Jordan, Cours d’analyse, ed. 2, vol. 2, p. 21.
20A. G. Greenhill, A chapter in the integral calculus (Francis Hodgson, 1888), p. 12: Differential and integral calculus, p. 399.
22The method sketched here is that followed by Stolz (see the references given on p. 21). Dr Bromwich’s method is different in detail but the same in principle.
23That the roots of are real has been proved already (p. 28) in a different manner.
24The superfluous root may be eliminated from the result by a trivial transformation, just as may be eliminated from
by writing this function in the form .
25Salmon, Higher plane curves, p. 29.
26Salmon, ibid., p. 29. French genre, German Geschlecht.
27We suppose in what follows that the singularities of the curve are all ordinary nodes. The necessary modifications when this is not the case are not difficult to make. An ordinary multiple point of order may be regarded as equivalent to ordinary double points. A curve of degree which has an ordinary multiple point of order , equivalent to ordinary double points, is therefore unicursal. The theory of higher plane curves abounds in puzzling particular cases which have to be fitted into the general theory by more or less obvious conventions, and to give a satisfactory account of a complicated compound singularity is sometimes by no means easy. In the investigation which follows we confine ourselves to the simplest case.
29See Niewenglowski’s Cours de géométrie analytique, vol. 2, p. 103. By way of illustration of the remark concerning particular cases in the footnote (§) to page 30, the reader may consider the example given by Niewenglowski in which
equations which appear to represent the straight line (part of the line only, if we consider only real values of ).
31This means of course that the equation obtained by substituting for and , in the equation of the line, their parametric expressions in terms of , has a repeated root. This property is possessed by the tangent at an ordinary point and by any line through a cusp, but not by any line through a node except the two tangents.
33I owe this remark to Mr A. B. Mayne. Dr Bromwich has however pointed out to me that substantially the same argument is given by Mr W. A. Houston, ‘Note on unicursal plane curves’, Messenger of mathematics, vol. 28, 1899, pp. 187–189.
34See Ptaszycki, ‘Extrait d’une lettre adressée à M. Hermite’, Bulletin des sciences mathématiques, ser. 2, vol. 12, 1888, pp. 262–270: Appell and Goursat, Théorie des fonctions algébriques, p. 245.
35If is the least common multiple of the denominators of the coefficients of powers of in , then
where is a polynomial. Applying Lemma (2), we see that , and so
36It is impossible that and should both vanish for , since is irreducible.
37‘Premier mémoire sur la détermination des intégrales dont la valeur est algébrique’, Journal de l’École Polytechnique, vol. 14, cahier 22, 1833, pp. 124–148; ‘Second mémoire…’, ibid., pp. 149–193.
38Raffy, ‘Sur les quadratures algébriques et logarithmiques’, Annales de l’École Normale, ser. 3, vol. 2, 1885, pp. 185–206.
39‘Mémoire sur les transcendantes elliptiques considérées comme fonctions de leur amplitude’, Journal de l’École Polytechnique, vol. 14, cahier 23, 1834, pp. 37–83. The proof may also be found in Bertrand’s Calcul intégral, p. 99.
40See Liouville’s memoir quoted on p. 45 (pp. 45 et seq.).
41The proof given by Laurent (Traité d’analyse, vol. 4, pp. 153 et seq.) appears at first sight to combine the advantages of both methods of proof, but unfortunately will not bear a closer examination.
42Second edition, vol. 1, pp. 267–269.
43‘Über diejenigen Curven, deren Coordinaten sich als elliptische Functionen eines Parameters darstellen lassen’, Journal für Mathematik, vol. 64, 1865, pp. 210–270.
44See Hermite, Cours d’analyse, pp. 422–425.
45See Legendre, Traité des fonctions elliptiques, vol. 1, chs. 26–27, 32–33; Bertrand, Calcul intégral, pp. 67 et seq.; and Enneper, Elliptische Funktionen, note 1, where abundant references are given.
46There is a similar theory for curves of deficiency 2, in which is of the sixth degree.
47See, e.g., Goursat, Cours d’analyse, ed. 2, vol. 1, pp. 257 et seq.
48See Hermite, Cours d’analyse, pp. 320 et seq.
49See Hermite, Cours d’analyse, pp. 352 et seq.
50See the remarks at the end of this paragraph.
51It is not difficult to give a purely algebraical proof on the lines of IV., §2.
52‘Mémoire sur l’intégration d’une classe de fonctions transcendantes’, Journal für Mathematik, vol. 13, 1835, pp. 93–118. Liouville shows how the integral, when of this form, may always be calculated by elementary methods.
53An interesting particular result is that the ‘error function’ is not an elementary function.
54The theorem with which Abel is engaged is a very much more general theorem.
55‘Or, au lieu de supposer ces coefficiens rationnels en , nous les supposerons rationnels en , ; car cette supposition permise simplifiera beaucoup le raisonnement’.
56Bertrand (Calcul intégral, ch. 5) replaces the last step in Abel’s argument by the observation that if and are both integrals of then is constant (cf. p. 39, bottom). It follows that the degree of the equation which defines can be decreased, which contradicts the hypothesis that it is irreducible.
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