ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 275 



analytique des Probabilites,' viz. that the integrals of differential func- 

 tions cannot contain other radical quantities than those that enter these 

 functions, theorems which were known to the first inventors of analysis 

 (see Leibnitz, ' Act. erud. Lips.'). In the first of the memoirs mentioned 

 above Liouville proposes the problem of finding the form of the integral 



\ydx, when this integral may be expressed algebraically, where y and x 



are connected by an algebraic equation 



y-Ly-i— . . , -My— ]Sr=0, 



L, M, . . . designating rational functions of a: ; he shows that the value 

 of such an integral is equal to a certain rational function of x and y. In 

 the discussion of this theorem he classifies functions of one or more 

 variables according to the irrationalities that enter them.' 



In the second memoir the general theorem which he proposes to 

 demonstrate is : if any algebraic explicit or implicit function y is given, 

 it is always possible to decide if it has or has not for an integral an 

 ■explicit or implicit algebraic function ; and if the question is decided in 



the affirmative, the same process will gis'o the value lydx. 



He shows that if the integral jydx may be expressed algebraically, it 

 has a value of the form : 



\ydx=a+l3y + y,f-+ . . . +X>/''-\ 



in which a, ft, y, . . . X are rational functions of x. 



In the twenty-third volume of the ' Journ. de I'Ecole Poly.' p. 37, 



Liouville finds that if the integral I ydx is expressible as an explicit finite 



function of x, it must be of the form : 



yydx=t + A logM + B log v+ . . . +C log IV, 



where A, B, . . . are constants, and t, «, v, . . . are algebraic functions of x. 

 This theorem is of course contained in the one of Abel in the precedino- 



article. Liouville further shows that if — ^, P and R denoting integral 



polynomials, cannot be expressed by an algebraic function of x, it cannot 

 be expressed as a finite explicit function of x ; from this follows that an 

 elliptic integral of either the first or second kind cannot be expressed as 

 an explicit function of its variable. (See also Liouville, ' Liouv. Journ.' 

 t. V. pp. 34 and 441, where it is proved that the same integrals, considered 

 as functions of their modulus, cannot be expressed in finite form.'-*) 



(36) Jacobi ('De functionibus duarura variabilium quadrupliciter 

 periodicis,' Crelle, bd. xiii. p. 55 ; ' Werke,' bd. ii. p. 25) proved that any 

 one-valued function of one variable cannot have more than two inde- 

 pendent periods, and that the ratio of these two periods cannot be a real 

 quantity and is irrational. 



' See also two memoirs by Liouville, ' Sur la Classification des Transcend antes ' 

 iLtouv. Journ. t. ii. p. 56, and t. iii. p. 523) ; and Poisson (Crelle, bd. xii. p. 89, and 

 bd. xiii. p. 93) 



' Cf. ZioKV. Journ. de I'JEcole Poly., t. siv. p. 137 ; and Ellis, p. 70. 



T 2 



