194 On the Expression for the roots of a complete Cubic, 



here propose to deduce from it the suitable expression when 



the equation is complete, as this is as likely to be useful as the 



other. 



Let the equation be 



x^ +pa,^ + gx + r = 0, 



then it is plain that we may accommodate the former expression 



to this case, provided we increase each of the roots of this by 



1 p^ 



- Pf and employ here S'— "^ for g' above; so that, still repre- 



senting a root by x^, the foregoing expression under the ra- 

 dical will now become 





{3x,^ + 2px,)+p^ 



A. -t 



and consequently the expression for the remaining roots is 



which may be stated in a rule, as follows : — 



Add «", to the coefficient of <2?'^, and call half the sum, with 

 changed sign, a. 



Subtract « from So^j, multiply the remainder by a. 



Subtract the coefficient of a; from the quotient, and call the 

 result /3. 



Then a+ \//3 will be the two remaining roots sought. 



Dr. Rutherford of Woolwich has recently published a neat 

 and ingenious method of attaining the above object when the 

 root ^, is developed by Horner's method, without contraction 

 of the decimals ; in which case it is remarkably easy when the 

 coefficient of a?'^ does not consist of many figures. The pre- 

 ceding rule involves nearly the same numerical woi'k whether 

 the coefficients be large or small, as x^ will usually have several 

 decimals ; and its application does not preclude a free abridge- 

 ment of Horner's columns. 



It is plain that, by a similar process, rules for the higher 

 equations, when all the roots but two are found, might be 

 contrived : but they would, in general, be complicated when 

 the equations are complete : when the second term of the 

 equation is absent, formulas for two roots, in terms of the 

 others, are given in the work before referred to. 



Belfast, Feb. 16, 1849. 



