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 

 b) be an imaginary root of unity, so that 



w" + w^ + 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 : 





+ V I Hi3 + V Hu 4- ^Hi5 4- V HiG 4- eV Hi7-VHi8^ 



4- V I Hi9 + V H20 4- B^ v^H2i4-Vh22 4- 9 l/a^s-Va^i} ; 



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

 that the twenty-four quantities. Hi, . . . 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 = Hi3 = H]9, H8= Hi4 =: H20, H9 = Hi5= H21, 

 Hio=Hi6=H22, HiiZ:Hi7 = H23, Hi2 = Hi8 = H245 H9 = Hii, Hio = Hi2. 



* Nuove Ricerche sulla Risoluzione Generale delle Equazioni Algebriche del 

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

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

 pendice" to the same work. 



VOL. II. 2 



