584 New Mode of' Resolving Algebraic Equations. 



2. If it be supposed that, in the roots of an equation of the 

 wth degree in z, /3„_i = 0, then, by taking the roots in a 

 proper order, we obtain, as before, 



(^[z) = n ^„_2, and, if ^ {z) = 0, ^^-2 = J 

 and, similarly, <^ (w) = w /3„_3, &c. 



3. Next, X being the root of the general equation of the wth 

 degree, let j/ = A^^ + M.r'*, then, in order that, in the equa- 

 tion in J/, /3,j_i may = 0, we have 



4^ (3^) = = A (f. {x"-) + M <f) (^^), .... (3.) 

 since <f) is a linear function. But ^ has many values arising 

 from the interchange of the roots one among another; let m 

 of the values of (3.) arising from this circumstance be multi- 

 plied together, and we have 



A'"^ + A"*-! Mtt' + A'^-^M^tt" + . . . M»'7r(»') = 0. (4..) 

 Now the peculiarity of the quantities w, tt', &c. is (see the 

 work above mentioned*), that one is derivable from another by 

 an easy process, and that when one consists of symmetric 

 functions of j:*, all do ; and if we select those forms of (^ which 

 are included in the expression u-^ -\- a!' (cf) {ii) — u^), giving r 

 every value from to w — 2, then, at least for the first four 

 degrees, tt is symmetric and (4.) becomes a homogeneous equa- 

 tion of the (»— l)th degree, whence -^ may be determined. 



4. We have thus obtained equations of the 2nd, 3rd and 

 4th degrees, whose roots are respectively of the forms 



a^, a + b^y a + b^ + c^, 



whence a certain convenient relation among the coefficients is 

 obtained (Mathemat. p. 83). 



5. To take away another term of the expression for the 

 roots, we must similarly assume z = A' j/^ + Myf* ; this gives 

 us, prima Jacie at least, the reduction of the biquadratic to 

 the binomial form, and of the equation of the 5th degree to 

 the solvable form of De Moivre, and may be found to throw 

 some light on the difficulties attending those transformations. 

 _ 6. The assumption indicated for taking away r terms of the 

 root, is 



Devereux Court, March 4, 1845. 



♦ The formal proof will appear in the next July number of that work. 



