REPORT ON CERTAIN BRANCHES OF ANALYSIS. 311 



functions of these roots ; and consequently, if irrational func- 

 tions of those roots are employed in the formation of the re- 

 ducing equation, the roots of the equation must enter into the 

 final expression of the required roots, in a form where that ir- 

 rationality has altogether disappeared *. If we assume, there- 

 fore, that such functions are in all cases rational, the next ques- 

 tion will be, whether they are discoverable in higher equations 

 than the fourth. 



This inquiry was undertaken by Paolo Ruffini, of Modena, 

 in his Teoria delle Equazione Algebraiche, published at Bo- 

 logna in 1799, and subsequently in the tenth volume of the 

 Memorie delta Societa Italiana, in a memoir on the impossibi- 

 lity of solving equations of higher degrees than the fourth. 

 He has demonstrated that the number of values of such func- 

 tions of the roots of an equation of w dimensions must be either 

 equal to 1 . 2 . 3 . . . w, or to some submultiple of it ; and that 

 when n = 5, there is no such function, the alternate function 

 being excluded, which possesses less than 5 values. The pro- 

 cess of reasoning which is employed by the author for this pur- 



* This proposition has been proved by Abel, in his Beiveis der Unmoglich- 

 keif algebraische Gleichungen von hoheren Graden als dem vierten allgemein 

 Aufzulosen, in the first volume of Crelle's Journal : the same demonstration 

 was printed at Paris, in a less perfectly developed form, during his residence 

 in that capital. This proof applies to algebraical solutions only, excluding 

 the consideration of the possibility of expressing such roots by the aid of un- 

 known transcendents. After defining the most general form of algebraical 

 functions of any assigned degree and order ; and after demonstrating the pro- 

 position referred to in the text, and analysing the demonstrations of Rufiini and 

 Cauch}% and showing their precise bearing upon the theory of the solution 

 of equations, he proceeds to show that the hypothesis of the existence of 

 such a solution in an equation of five dimensions will necessarily lead to an 

 equation, one member of which has 120 values and the other only 10 ; an ab- 

 surd conclusion. It is quite impossible to exhibit this demonstration in 

 a very abridged form so as to make it intelligible ; and though some parts of 

 it are obscure and not perfectly conclusive, yet it is, perhaps, as satisfactory, 

 upon the whole, as the nature of the subject will allow us to expect. 



It is impossible to mention the name of M. Abel in connexion with this 

 subject, without expressing our sense of the great loss which the mathematical 

 sciences have sustained by his death. Like other ardent young men, he com- 

 menced his career in analysis by attempting the general solution of an equa- 

 tion of five dimensions, and was for some time seduced by glimpses of an 

 imagined success ; but he nobly compensated for his error by furnishing the 

 most able sketch of a demonstration of its impossibility which has hitherto 

 appeared. His subsequent discoveries in the theory of elliptic functions, 

 which were almost simultaneous with those of Jacobi, have contributed most 

 materially to change the whole aspect of one of the most difficult branches of 

 analytical science, and furnish everywhere proofs of a most vigorous and in- 

 ventive genius. He died of consumption, at Christiania in Norway, in 1827, 

 in the 27th year of his age. 



