252 REPORT— 1897. 



be expressed by one such integral, where the upper limit of this integral 

 is a rational function of the upper limits of the other integrals. Similar 

 results are found for the elliptic integrals of the second and of the third 

 kinds. For those of the second kind there enters, in addition, an algebraic 

 function, and for those of the third kind a logarithmic algebraic function. 



Abel considered the integrals of any algebraic functions, and established 

 a theorem for the transcendents that arise from the integrals of these 

 functions, which has for them the same meaning as Euler's theorem has 

 for the elliptic transcendents. 



The question proposed by Abel is : Suppose X in formula (I.) above is 

 any algebraic function of x, then is it possible, taking different variables, 

 to establish algebraic (or logarithmic) relations between integrals of the 

 form 



F dx 



"<^'=i:^' 



when the variables are connected by requisite algebraic equations ; that is, 

 can algebraic (or logarithmic) relations be found among 



U(xi), n(a;2), , . . n(a;„), 



when x^, X2, . . . x^ are connected by algebraic equations 1 If such is 

 the case, the question next arises : How many algebraic equations are 

 necessary, and do these equations depend upon the nature of the 

 function X ? 



Abel, in his celebrated paper, 'Memoire sur une Propri^t^ Gt^nt^rale d'une 

 Classe Tres-Etendue de Fonctions Transcendantes,' 'CEuvres Completes,' 

 t. i. p. 145 (Sylow and Lie), considered the question in a still more 

 general form, and found that all those functions whose derivatives may be 

 expressed through algebraic equations, in which the coefficients are 

 rational functions of one and the same variable possess properties that 

 are analogous to those of the elliptic functions stated above. 



The results of these investigations are expressed in the following 

 theorem, known as Abel's theorem : If we have several /unctions whose 

 ■derivatives may he (expj'essed as) the roots of one and the same algebraic 

 equation, and all the coefficients in this algebraic equation are rational 

 /unctions of one and the same variable, then it is always possible to express 

 the sum, of any number of functions which are like the first functions by 

 ineans of an algebraic {and logarithmic) function, provided a certain 

 number of algebraic relations can be established between the variables of the 

 ficnction in question.^ 



The number of these relations does not depend upon the number of the 



dw, dm dm„ 



» Such functions are ^ = R (y„a;,), -^' = R (.y^x^), ... ^ = R (yn.*™). 



■R denoting a rational function, where Xi(i = 1,2,. . . n) are the points of inter- 

 section of two curves x^^'iV) = ^^^ ^ (^^V) = 0. a-nd j/i are the corresponding 

 values of y that are obtained from these two equations. 



Now every symmetric function of the solutions common to x C"'. V) = 0> ^.nd 

 '6(x,y) = is a rational function of the coefficients of these two equations. 



Hence 2 I 'R(x,y')dx is an one-valued function of the coefficients of x(*. V) = 0, 



C=lJ 



