318 THIRD REPORT — 1833. 



framed * ; and it arises from the application of such rules that 

 we are enabled to determine the coefficients of an equation of 

 which those periods are the roots, and thus to depress the 

 original binomial equation to one whose dimensions are the 

 greatest prime number, which is a divisor of w — 1. 



It follows, therefore, that if the highest prime factor of » — 1 

 be 2, the resolution of the binomial equation a;" — 1 = will 

 be made to depend upon the solution of quadratic equations 

 only, and consequently to depend upon constructions which 

 can be effected by combinations of straight lines and circles, 

 and therefore within the strict province of plane geometry : 

 this will take place whenever n is equal to 2* + 1 and is also 

 a prime number. Thus, if A' = 4 we have « = 17, a prime 

 number, and therefore the solution of the equation a:'^ — 1 = 

 will be reducible to that of four quadratic equations. Similar 

 observations apply to the equations 



^2V 1 _ 1 = and x'^^^+ 1-1=0. 



The same principles which enable us to solve algebraically 

 binomial equations, under the circumstances above noticed, will 

 admit of extension to other classes of equations, whose roots 

 admit of analogous relations amongst each other. Gauss f has 

 stated that the principles of his theory were applicable to func- 



tions dependent upon the transcendent /^yj\ 4\> which de- 



fines the arcs of the lemniscata, as well as to various species of 

 congruencies ; and he has also partially applied them to certain 

 classes of equations dependent upon angular sections, though 

 in a form which is very imperfectly and very obscurely deve- 

 loped. Abel, however, in a memoir X which is remarkable for 

 the generality of its views and for its minute and careful ana- 

 lysis, has not merely completed Gauss's theory, but made most 

 important additions to it, particularly in the solution of exten- 

 sive classes of equations which present themselves in the theory 

 of elliptic transcendents §. Thus he has given the complete 



* Symmetrical functions of these periods will be multiples of the sum ( — 1) 

 of these periods and of 1. This conclusion follows immediately from the re- 

 placement of the arithmetical by the geometrical series of indices, according 

 to the general process of Lagrange, without any antecedent distribution of 

 the roots into periods. See Note xiv. to the Resolution des Equations Num.e- 

 riques. It follows from thence that the coefficients of the reducing equations 

 will be whole numbers. 



■f Disqiiisitiones Arithmetic(B, pp. 595, 645. 



I " Sur une Classe particuliere d'Equations resolubles algebriquement," — 

 Crelle's Journal, vol. iv. p. 131. 



§ Crelle's Journal, vol. iv. p. 314, and elsewhere. 



