IV. On the Resolution of Algebraical Equations. 

 By R. MURPHY, B.A. 



FELLOW OF CAIUS COLLEGE; AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. 



[Read March 7, 1831.] 



INTRODUCTION. 



The researches of Lagrange on that part of Pure Analysis, 

 which forms the subject of the present Memoir, have been fol- 

 lowed up with considerable success by many foreign Mathe- 

 maticians, amongst whom M. Augustin Cauchy deserves to 

 be particularly distinguished ; indeed the extensive use of the 

 theorem of Lagrange in Physical Astronomy, had turned the 

 attention of Analysts to consider more intimately the nature of 

 that series, the conditions of its convergence, the root which it 

 particularly represents, &c. I have referred to as many papers 

 on this subject, scattered through the Memoirs of the French 

 Institute, the Journal of the Polytechnic School, and the Annales 

 de Mathematiques, &c. as I conveniently could. I do not find 

 that in the point of view in which this subject is here exhibited, 

 I have been anticipated in any of the articles above referred 

 to, a point on which it is necessary to be doubtful, without 

 actual reference, from the great number of persons who have 

 been recently, and are at present, engaged in extending the limits 

 of Analysis. 



