275 



August 4. 

 REV. C. W. WALL, D. D., in the Chair. 



The President made a communication respecting a me- 

 thod which had been lately proposed by Professor Badano 

 of Genoa, for the solution of algebraical equations of the 

 fifth and higher degrees.* 



Lagrange has shown that the function 



receives only twenty-four different values, for all possible 

 changes of arrangement of the five quantities, x', . . . x^, if 

 u> be an imaginary root of unity, so that 



w' -1- w^* -f w^ + w -f 1 = 0. 



Professor Badano has proposed to express these twenty- 

 four values by certain combinations of quadratic and cubic 

 radicals, suggested by the theory of biquadratic equations, 

 and having the following for their type : 



<^ = Hi -f- Vh2+-^H3-|--v/h4-|- V«5- Vhs 



+ V I Hy -f V Hs + ^lh+ V Hio -f >C^Hii — Van} 



4- V J Hi9 -I- VH20 + 0' ^H2i+ V H22 + ^H23- V H24 1 ; 



9 being here an imaginary cube root of unity. He contends 

 that the twenty-four quantities, h,, . . . H24, are all symme- 

 tric functions of the five quantities x',...x^; and that they 

 are connected among themselves by the sixteen relations 



H3=H5, H4 = He, H7 = H13 = Hi9, H8=: Hi4 = H20, H9 = His^Hai, 

 Hio=Hi6=H22, HiiI=Hi7 = H23, Hi2=Hi8=H24, H9=Hii, H,o=Hi2. 



• Nuove Ricerche sulla Risoluzione Generale delle Equazioni Algebriche del 

 P. Gerolamo Badano, Carmelitano scalzo, Professore di Matematica nella R. 

 Universita di Genova. Genova, Tipografia Ponthenier, 1840. See also an " Ap- 

 pendice" to the same work. 



VOL. II. 2 



