518 Mr. R. C. Rowe. Memoir on Abel's Theorem. [June 10, 



ascending powers of each simple factor x — a'va. the denominator of 



0(.u), taking in each development the coefficient of — — , adding to- 



x — a 



gether the coefficients thus obtained, and subtracting from the result 



the coefficient of - in the development of the same function (p(x) f(x) 



x 



in descending powers of a?."* 



Then the theorem of this section is 



2 !>(,, y)dx=e [ 1 1 f log e(y) + 0. 

 J L/s(a>)Bo(*)J x(y) 



This formula is more general than either of Boole's, while it seems 

 more concise and intelligible than that of Abel to which it corre- 

 sponds. 



The rest of this section is occupied with the application of the 

 theorem to several simple cases. Those have been chosen of which 

 the results were well known (e.g., the circular and elliptic f unctions), 

 with a view to the comparison of method. 



In the second section it is shown to follow from the results of 

 Section I that the sum of any .number of integrals of the form con- 

 sidered may be expressed in terms of a number of such integrals equal 

 to the number of conditions necessarily connecting the variables in 

 the proposition already discussed ; and the question is investigated. 



What is the least value of which this number of conditions is susceptible ? 



It is proved independent of everything but the form of the 

 equation y) = by which y is determined as a function of x ; the 



m i 



answer being that if this equation has n^ roots of the form y — CA, 

 n 2 /io of the form CxjT^ and so on, then this least number is 

 !Zn r mrn s fis + ^rfimfi — \Smm — — \ n + 1. 



s>r 



The expressions for the roots are according to descending powers of 



x ; each 7 — is a fraction in its lowest terms with denominator positive ; 

 fi 



the series — l , — , etc., are in descending order of magnitude ; and this 



equality of powers in sets is shown to subsist. 



This section concludes with an example of its processes. 



The third section contains three distinct parts : first, a generalization 

 of the theorem of Section I, showing that a similar expression to that 



* Cauchy had used in his " Calcul des Kesidus " a symbol which only differs from 

 this in the absence of the subtractiye term last mentioned. 



