Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/82

72 INFINITESIMAL CALCULUS tioned, the chief is the unfinished Precis d une theorie des fractions elliptiques, " which appeared in Crclle, iv. , in 1829. " The whole of my researches will form a work of some extent which 1 cannot yet publish, therefore I give here a Precis of the method I followed, and its general results." The fragment of this work which has been published deals only with the integrals. 260. The consideration of the indeterminateness of the integral /*: f n dx which gives rise to periodicity in the inverse function o? = sin amw, has led to the consideration of the whole subject from a new point of view. The introduction of the complex variable into analysis by Cauchy in his Memoire sur les integrates definies prises entrc des limitcs imaginaircs (1825), and by Gauss in the second part of his Theoria Residuorum Biquadraticorum (1831), has been followed by the works of Puiseux ( Recherchcs sur les fonctions algebriques," Liouville, xv. 1850), of Riemaun (Inaugural Disser tation, 1851, and "Theorie der Abelschen Functionen," Crelle, 1857), and of Weierstrass ("Theorie der Abelschen Functionen," Crelle, 1856) which develop the subject in this more extended field, perfecting the conception which the term function covers in analysis, and pointing out the essential distinctions in the different modes of dependence of two quantities, such distinctions, for instance, as when a function is defined by a differential equation, whether it is one-valued or not, and, if it be, whether it is integer or frac tional. 261. In close connexion with this is another department to which the theory of transcendents has with great success been applied, the investigation of the geometrical properties of curves. The points on a curve are expressed as functions of a parameter, and on the nature of these functions the nature of the curve depends, the "deficiency" or " Geschlecht " of the curve (see CURVE, vol vi., p. 725) determines the nature of the function, and any curve into which another can be rationally transformed depends on the same function. We shall conclude with a brief application to the case of elliptic functions and plane curves of the third degree. It is well known that the equation of any non-singular cubic can be reduced to the form yz 2 = x(x-y)(x-k 2 y) , where y = is the tangent at the point of inflexion in which the curve meets = 0, and x = 0, x = y, x=k 2 y are the tangents from that point to the curve, their points of contact lying on z 0. This equation is satisfied identically by assuming the equations px = sin am u, py = sin 3 am u, p2 = Aam u cos ami*, which determine any point on the curve by a parameter u. To each value of u corresponds a perfectly definite point of the curve. But on the other hand, to any point of the curve corresponds an infinite number of values of the argument all related to one of them, u, differing from it only by a multiple sum of the periods. The occurrence of the elliptic integral u here in this normal form results from the coordinates chosen ; but, whatever they be, we see that the points of the curve can be expressed by a parameter depending on no higher irrationality than that we have intro duced. When the cubic has a double point, the coordinates of any point on the curve can be expressed by a parameter without intro ducing any irrationality. 262. To investigate the intersections of the cubic with a right line we proceed to derive in a simple manner a slight extension of Euler s integral ( 207). Written in Jacobi s notation it is which is easily thrown into the form + k cos am *i 1 cos am w 2 cos am(M 1 = Aam Aam This may be extended to three arguments as follows. Denoting sin am w r briefly by s r , also cos am u r by c r , Aarn v r by A r , tan am u r by t r , and cot am u r by d r , the formula may be written & 2 + 2 c 1 c 2 c(!{ 1 + ) = A 1 A 2 A(z< 1 + M. ! ) ; putting for w 2 , u. 2 + u,^, and expressing, by 207, c( 2 + M 3 ) and A (MO + u 3 ) by functions of one argument, we get k - + k-c^c^u^ + u. 2 + u 3 ) - A! A 2 A 3 A( u-^ + u. 2 + U 3 ) = k-s. 2 s 3 {k -s. 2 s 3 + c l A. 2 ^c(u 1 + u. 2 + u, A ) - A 1 c 2 c 3 A(! 1 + 2 + M 3 )} . Now the former expression is symmetrical ; denoting it by ks.^s 3 9, we can determine as follows. Writing for brevity c(zt 1 + M 2 + M 3 )= 0, and A(u 1 + u. 2 + u 3 } = A. , the equation is k ~s. 2 s s + CjA.jA.jC - AjC/jjA - 9s 1 = . Hence writing down the three equations, which must hold from symmetry, = k -s. 2 ft 3 + CjA-AjC - AjC.jC.jA - ft?! , =^ 2 s/! + r 2 A 3 A 1 C - AjjCgCjA - 6s. 2 , = /fc 2 5jS.j + c 3 AjA.,C - A^CjC-jA - 6s 3 , we obtain C and A as quotients of determinants of single argu ments. For A we get 1 <1 ^ 2 2 AT ] S , 2 ^ 3 A 3 and for C i ^1^1 1 ^i" ^i 7 1 S 3 2 A.j< 3 But in this, increasing each argument by iK , since then we get, fors^-r- , for c r , -7; -, for A r , -ict r , for t,. , , and thus iA for C, rn . where S stands for sin am(M 1 + u. 2 + u 3 ), this formula gives A.AoA, 1 p 2 A r t -L t>i iiiO(/i 1 s 2 2 A 2 c< 2 I S 3 2 A 3 C< 3 whence Si ^ 1 c 2 C -2 S 2 1 , ^ AC and the value for 6 thus found gives + -9 + %) ~ A 1 A 2 A 3 A(j< 1 + M 2 + u s ) 263. The formula thus obtained for sin am(! + M 2 + M 3 ) vanishes when ? 1 + ?6 2 + it 3 = 0, or differs from only by an integer com bination of the periods. But the determinant vanishes if its constituents be the coordinates of three collinear points. But these are, as we have just seen, the coordinates of three points on the cubic yz 2 =x(x i/)(x-k 2 y), 261. This result may therefore be stated thus : If the points of a cubic be expressed as elliptic functions of a parameter, then for the inter sections ivith a right line the sum of the arguments differs from zero only by some integer combination of the periods. This enables us to solve many problems. For instance, the argu ments of the points of contact of the four tangents which can be drawn to the curve from a point u on it are u + ta U + ia + <a Conversely the tangential point u of a given point v of the curve is determined by u E; - Iv (mod ea, u). The problem of determination of points of inflexion when one point of inflexion is known is identical with the problem of the special trisection of elliptic functions, i.e., of the determination of the values for u = t o to the same moduli. Bibliography. In addition to tho works on elliptic functions and the higher transcendents already named, there have recently appeared as independent works, besides innumerable memoirs in the various mathematical periodicals Briot and Bouquet, Theorie des fonctions elliptiqites, 2d ed., 1375; Briot, Theorie des fonctions abeliennes, 1879 ; Booth, Theory of Elliptic Integrals, 1851 ; Casorati, Teorica delle funzionidi variabili complesse, 18(58; Cayley, Elementary Treatise on Elliptic Functions, 1876; Clcbsch, Geo/netrie,IS~6; ClebschandGordan, Theorie der Abelschen Functionen, 1866 ; Durege, Elemente der Theorie der Functionen, 1873 ; Id., Theorie der ellipt, Funct., 1878 ; Eisenstein, Ueitriige," collected in his Math. Abhandlungen, 1847; Ellis, " Report on Recent Progress of Analysis," Brit. Ass. Reports, 184R ; Enneper, EU. Funct. Theorie und Geschichte, 1876 ; Kb niRsherger, Transformation, Ac., der EH. Functionen, 1868; Id., Theorie d. EU. Funct., 1874; Id., Theorie d. hyperelliptischen Integrate, 1878; Id., Getchichte der Ell. Transcendenten, 1879 (to the last two authors we arc mainly indebted for the historical details of our subject); Lipschitz, Differential und Integralrechnun/j, 1880; Neumann, Ucber Riemann s Theorie d. Abelschen Integrate, 1865; Roberts, On Addition of Elliptic and Hyperelliptic Integrals, 1871 ; Russell, " Report on Recent Progress in Elliptic and Hyperelliptic Functions," Brit. Ass. Reports (1869, p. 334; 1870, p. 102; 1872, p. 335, 1873, p. 307) ; Schcllbach, Ell. Integralen und Theta Functionen, 1864 ; Schlomilch, Compendium der Hoheren Analysis, 2d ed., 1874 ; Sohncke, " Elliptische Functionen," in Ersch and Gruber s Ency- clopddie; Thomae, Abriss einei- Theorie der complexen Functionen, 1873; Id., Sttmmlung von Formeln,1ST6; Verhulst, Traite elementaire des fonctions ellip- tiques, 1841 ; Weber, Theorie d. Abelschen Functionen rom Geschlecht 3, 1876; Veyr, T forie d. Ell. Fnnct.. 1876 ; and of the highest historical interest is the pub lication by Borcliardt of the correspondence between Lependre and Jacobi in vol. Ixxx. of Crelte s Journal, 1875, reproduced in vol. i. of the collected works of Jacobi, 1881. (15. W.)