298 THIRD REPORT— 1833. 



following problem, and examine all the consequences to which 

 its solution will lead. 



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

 shall be equal to^^,, 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/>„." 



The quantities x, x^, . . . *•„_!, 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 9 + -v^ — 1 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 pi, p^^, . . . pn'- 

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

 and Pj, Pg, . . . P„_i to be the quantities corresponding to />i, 

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

 we shall get 



X + Pi =^,, 



ar Pi + Pa = Ihy 



a: Pg + Pg = /?3, 



a?P„_2 + P„-l = pn-U 

 X P„-i = Pn- 



If we multiply these equations from the first downwards by the 

 terms of the series ^""^ 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^—ih *'""' + Ih '^""' -... + (—!)"/>„ = 0. (1.) 



In as much as p^, jt?^, . . . pn 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 mvist compre- 

 hend the expression of all the roots. 



By this mode of presenting the question we are authorized in 

 considering the syinholical composition of the coefficients of 

 every 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- 



