with some Supplementary Remarks on Newton's Rule, 125 



I have justified the statement in the paragraph here alluded 

 to by a reference to the series of limiting cubics ; but the truth 

 of it may be easily established in a direct manner without any 

 such reference — thus : — 



The group of conditions given in my paper in the Journal for 

 August 1865 are all comprehended in the general form 



(m + l)(n—m — l)A m _!A m+1 > m(n— m)A? m , . . (1) 

 where m is the exponent of x in the middle one of any triad of 

 terms. That is to say, the triad itself being 



A m+1 ^ +1 + A m x m + A m _, x m ~\ 

 the triad derived from this is 



and the principle affirmed is, that accordingly as (1) holds or 

 fails, so will the following condition, in which this derived triad 

 replaces the former, hold or fail, namely the condition 



m{n— m + 2)(m — l)(m + l)A m _ 1 A TO+1 > (?/i-l)(w-m-l)(fflA m ) 2 , 



or, expunging the factors common to both sides, the condition 



(n— m + 2)(m + l)A m _ 1 A m+l > m(n — m — 1) A 2 TO . 



Now it being remembered that, for the triad with which we are 

 dealing, the degree of the equation is n— 1, and not n, (as in the 

 case of the former triad) we must put n — 1 for n in this last con- 

 dition : the form will then be 



((w-l)-m + 2)(m + l)A m _ 1 A w+1 >m((n-l)-w-l)A 2 wz , 



which is the same as 



(m + l)(n — m — 1) A w _ 1 A fn+1 > m(n — m)A: 2 mi 



and is thus identical with (1). 



I may add, in conclusion, that if any number p of consecutive 

 triads fulfil the condition of equality, then there will always be 

 at least p + 1 or p imaginary roots ; the former number if p be 

 odd, and the latter if p be even. 



The truth of this is readily seen. If the derived equation, of 

 which the first member terminates with the last of these triads, 

 be replaced by its reciprocal, we shall have p consecutive leading 

 triads fulfilling the condition ; and by diminishing or increasing 

 the roots of this reciprocal equation by the number which causes 

 the second term of the transformed equation to be zero, the next 

 p terms will be zero also, just the same as if all the following 

 triads in the equation fulfilled the condition of equality. And 

 distinct sequences of this kind imply distinct groups of imagi- 

 nary roots. 



June 13, 1866. 



