373 



[«', which, embracing two roots &ff(x) = 0, embraces also the root 

 of the middle equation of the first degree.* It thus appears that those 

 imaginary roots of an equation which hare not, at first, the character 

 of what we have called primary pairs, become convertible into primary 

 pairs by the same process of continuous approximation by which they 

 would be separated and computed if they were real. And we submit 

 that all desirable extension and efficiency is thus given to Horner's 

 method of development, and to the general criteria at page 344. 



(32) As to the practical operation of carrying on the development 

 adverted to, when two or more roots are indicated in the same interval, 

 and are Ion? in separating when real, or in disclosing their character 

 when imaginary, we must refer for the necessary directions — more 

 especially as to the trial divisors for facilitating the discovery of the 

 successive figures of the real root actually approximated to, to [* The 

 Theory and Solution of Equations of the Higher Orders," pp. 259-263. 

 But we may add here that, in testing a triad of coefficients by the con- 

 dition of imaginarity, if the figures of these coefficients are numerous, 

 and it be seen necessary to compute with some degree of precision, the 

 squaring and multiplying may be tedious operations. In such cases we 

 would recommend a shorter method of proceeding, thus : — Let the pro- 

 duct, with its proper ^Newtonian multiplier, be represented by pPP', 

 and the square, with its proper multiplier, by gQ 2 : then the condition 



P PP >^impHes^>i| 



and these division operations being carried on, a figure at a time alter- 

 nately, we shall rind which quotient exceeds the other without com- 

 puting even a single superfluous figure. 



(33 ^e shall tenninate this paper with the investigation of a 

 general formula for determiriing the character of the roots of a complete 

 cubic equation, independently of actual development. 



If either of the two triads of a cubic equation satisfy the condition 

 of imaginarity, no special formula for this purpose will be necessary : 

 we have therefore only to provi le foi the case in which both triads fail, 

 or in which the square of the middle term of each (with its proper fac- 

 tor), minus the product of the extremes (with its proper factor), is a 

 positive quantity. Let 



P = A 3 a? -f + A x x + A 0 ; or, page 367, X 0 

 Q = 3A 3 x s + 2A,x + A 1 „ 



v 3PP' = AJ - 3A : Aiz- + A- A, - 9J c ^ 3 >+(^i 2 - 3^c4 2 ) . . [1 ] 



* The passage of this root being attended with the loss of two variations ; that is. 

 variations are lost in passing from r - I to r — I. 



