Eoss — Verb-Funetions. 



65 



we can obtain a derived equation in 2 which, when treated by 

 descending diyision, will give the same results ; so that we need only 

 consider formulae for descending division. Thus the original equation 

 can be attacked in 2m ways — as already suggested in § 15 (3). 

 Taking the biquadratic equation for example, we have 



^- ao!^ ^ hx^ -V cx ^ = - d, 

 x^ ax^ -\- hx + dx'^ = - ^, 

 X' ax ■\- + cx~'^ + dx~- = - h, 

 X -\- ^ Ix''^ 4- cx'^ + dx~^ = - a; 



and by putting x = we obtain a derived equation in z which has 

 four similar forms — eight altogether. 



(3) But on examining the 2m inverts derived from these forms, we 

 shall find that most of them are either arithmetically unintelligible, or 

 have unreal subjects. Hence, in order to save labour, we must seek a 

 method for quickly detecting which forms will yield useful results — 

 that is, in the example just given, which of the subjects, 



- d, — c, - h, -a, 



may be employed. 



Eeverting to the general equation 



we obtain, by § 17, 



where g = "J- p.n, and denotes the operation of giving the proper 

 coefficients to the terms. jS'ow this expression for x consists of a 

 number of fractions raised to all possible positive integral powers and 

 often combined with each other in various ways. If one of the 

 fractions be greater than unity, the expression for x will contain a 

 certain number of terms which, if for the moment we neglect the 

 effect of 0, will tend to infinity. Hence for arithmetical purposes, 

 neglecting the effect of 0, all the said fractions must be less than 

 unity. That is, if P_a be any one of the original coefficients, 



must be less tlian unity; that is, "J- must be greater than 



9 



E 2 



