EjOSS — Verh-Fimcfions. 



49 



each give one root by itself ; and the forms 



1 o ^ , 1 ^ 



X ^ - x^ = - - and ^ -x =^ - ^ 

 a a 



give the other by itself — the first root being obtained by descending 

 division, and the second by ascending division. 



(7) The cubic x^+ ax -^h = 0. — The complete solution is given by 



11 V 3 y [6^ 



the coefficients of the integral powers of ^ vanishing. This does not 

 coincide with the expansion of the sum of two cubic roots obtained by 

 Cardan's solution, because that has nine values, and is not the exact 

 invert. When the two cube roots are properly taken however, their 

 expansion agrees with the value of 



obtained by operative division; and, by § 14, 



Dividing the original equation by x, x"^, and x^^ we obtain subsi- 

 diary equations which give single roots when the first term of the 

 divisor is P or ^"^ : one root by ascending division, and one by 

 descending. Two roots are obtained when /8- or is the first term 

 of the divisor. 



(8) Find solutions for 



X + ax^ -^^ hx^ . . . = and x + a -\- hx~'^ + cx'"^ . . . = y, 



the number of terms being indefinite. The inverts of 



/8 + + hl3K.., and of fS + -t + c^" . . . 



can be found without difficulty by the same process ; and the exercise 

 will be instructive. As the first term in both cases is y8, the inverts 

 will be free from radicles, and will consequently have only one value 

 each. The first form may be used for any equation of which the 



D 2 



