^^S THIRD REPORT — 1833. 



following problem, and examine all the consequences to which 

 its solution will lead. 



" To find n quantities x, x^, x^, . . . x„_i, such that their sum 

 shall be equal to ^9,, the sum of all their products two and two 

 shall be equal to p^, the sum of all their products three and 

 three shall be equal to p^, and so on, until we arrive at their 

 continued product, which shall be equal to p„." 



The quantities x, a-j, . . . x„^i, are supposed to be any quan- 

 tities whatever, whether real or affected by any signs of affec- 

 tion whether known or unknown. It is our object to show that 

 the only sign of affection required is cos 6 + -/^ sin 9, taken 

 in its most general sense. 



It is very easy to show that the solution of this problem will 

 lead to a general equation, whose coefficients are p^, p^, . . . jo„ : 

 for if we suppose the first of these quantities x to be omitted, 

 and Pj, Pg, . . . P„_i to be the quantities corresponding to p^, 

 pc^, . . . pn when there are (w — 1) quantities instead of n, then 

 we shall get 



a; + Pj =/?„ 

 a: Pi + Pa = p<i, 



XV^ + P3=^3, 



ea^ 



^ P»-2 + P»-l — pn-l> 

 ^ Pn-1 = Pn- 



If we multiply these equations from the first downwards by the 

 terms of the series x"-', x"-^, . . . x'^, x, 1, and add the first, 

 third, fifth, &c., of the results together, and subtract the second, 

 fourth, sixth, &c., we shall get the general equation 



x"~23i a:"-' + j)^ x"-^ -... + (- 1)«^„ = 0. (1.) 



In as much as />,, p^, . . . p„ may represent any real magni- 

 tudes whatever, zero included, it is obvious that we may consi- 

 der this equation as the result of the solution of the problem in 

 its most general form. And in as much as x may represent any 

 one of the n quantities involved in the problem, we must equally 

 obtain the same equation for all those n quantities : it also fol- 

 lows that every general solution of this equation must compre- 

 hend the expression of all the roots. 



By this mode of presenting the question we are authorized in 

 considering the spnbolical composition of the coefficients of 

 everi/ equation as known, though the ultimate symbolical form 

 of the roots is not knoivn ; and our inquiry will now be properly 

 limited to the question of ascertaining whether symbols repre- 



