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XIV. — On Equations of the Fifth Degree : and especially on a certain System 

 of Expressions connected with those Equations, which Professor Badano* 

 has lately proposed. By Sir William Rowan Hamilton, LL.D., P.R.I.A., 

 F.R.A.S., Honorary I/ember of the Royal Societies of Edinburgh and 

 Dublin ; Honorary or Corresponding Member of the Royal or Imperial 

 Academies of St. Petersburgh, Berlin, and Turin, of the American Society 

 of Arts and Sciences, and of other Scientific Societies at home and abroad ; 

 Andrews' Professor of Astronomy in the University of Dublin, and Royal 

 Astronomer of Ireland. 



Eead 4th August, 1842. 



1. JLAGRANGE has shown that if a be a given root of the equation 



a"-' -I- a"-* + . . . + a* -f a+ 1 ^ 0, 

 n being a prime factor of m, and if n denote for abridgment the quotient 



1.2.3. ..m 

 (1.2.3...-) 



then the function 



t = x' -{- ax" -{- a^x'" + . . . + a^-'ar'"" 



has only jj. different values, corresponding to all possible changes of arrangement 

 of the m quantities a/, x", ... jr'"", which may be considered as the roots of a 

 given equation of the m"* degree, 



^■» _ Aar^-' + Bar*"-*— c^"-^ -j- . . . = ; 



• Nuove Ricerche sulla Risoluzione Generale delle Equazioni Algebriche del P. Gebolamo 

 Badano, Carmelitano scalzo, Professore di Matematica nella R. Universita di Geneva. Geneva, 

 Tipografia Ponthenier, 1840. 



VOL. XIX. 2 U 



