ON THE HISTORICAL DEVELOPMENT OK ABELIAN FUNCTIONS. 253 



functions, but only upon the nature of the particular functions that ar© 

 considered. 



The same theorem is still true when we suppose the functions multi- 

 plied by any rational number positive or negative. 



We may therefore deduce the following theorem : Wc are always able 

 to express the sum of a given number of functions, wliich are multiplied 

 each by a rational number, and in which the variables are arbitrary, by a 

 similar sum of functions, Avhose number is determinate, and in which the 

 variables are algebraic functions of the given functions. 



As a further consequence is the theorem that the sum of any number 

 of integrals of the form considered may be expressed by the sum of a 

 definite number of such integrals with (perhaps) the addition of a deter- 

 minate algebraic (and logarithmic) expression, in which the variables are 

 algebraic functions of the variables of the first integrals. 



We therefore have the following result : Although in general we cannot 

 integrate an algebraic function by means of algebraic or logarithmic 

 functions, we may, however, obtain for the sum of a certain number of 

 such transcendental integrals an expression which is composed of algebraic 

 (and logarithmic) functions. 



Abel considered further the smallest number p of integrals through 

 which the sum of any number of other integrals may be expressed. This 

 is the well-known number which denotes the class (Classen- zahl) of the 

 connectivity of Riemann's surface xi^t 2/)> upon which y is an one- valued 

 function of x, and Clebsch's deficiency of the algebraic cui've x(^"j y) = 0. 

 (See Cayley's ' Addition to Mr. Rowe's Memoir,' loc. cit. p. 752.) 



Jacobi (' Werke,' bd. i. p. 379) writes : ' To this theorem we prefer to 

 give as the most beautiful monument of [Abel's] extraordinary intellect, 

 the name Abel's theorem, since it bears the entire stamp of his depth of 

 thought. We consider it the greatest mathematical discovery of our time, 

 as it in a simple form, and without the apparatus of the calculus, gives 

 utterance to the deepest mathematical thought.' 



Legendre calls the theorem a monumentum cere perennius (letter to 

 Jacobi, Jacobi's ' Werke,' bd. i. p. 376). 



The theorem is contained in a paper written in the year 1825, but not 

 published until after Abel's death : ' Sur la comparaison des fonctions tran- 

 scendantes ' (' CEuvres,' t. ii. p. 55). The theorem so stated in this paper is : 

 The sum of any number of functions which have an. algebraic differential 

 may be expressed through a definite number of such functions. It is de- 

 veloped in the large memoir above mentioned, ' Memoire sur une propridte, 

 etc.,' which was presented to the French Academy, October 1826, and not 

 published until 1841 in the 'Memoires des Savants Etrangers,' t. vii. 



Legendre in the third supplement of the ' Traits des fonctions ellip- 

 tiques,' p. 191, gives to the transcendent 



and e(x, y) = 0, this one-valued function being by Abel's theorem an algebraic (and 

 logarithmic) function. 



The points {Xi,yi') (i = 1, 2, . . . ?^) are not independent of each other, but as 

 soon as a certain number of them is given, the remaining jy (say) are of themselves 

 determined, being the roots of an algebraic equation of the ^th degree, whose 

 coefficients are rational functions of those points that are given, so that between 

 these coefficients there exist p algebraic relations. 



