364 



coefficient of the first term, be not a complete power (a n ), the root of 

 which is a = - A n _ x -j- nA m we should at once know that the poly- 

 nomial cannot be the development of any expression of the form 

 A n {x ± a) n ; nor can it be if the terms be neither all positive, nor yet alter- 

 nately positive and negative. Yet if any number of consecutive triads 

 satisfy the conditions of equality, an expression of this form may 

 always be found by computing towards the left, as well as towards the 

 right (if the consecutive triads be intermediate terms), such that those 

 terms shall be identical with the corresponding terms of the develop- 

 ment for some values of A n and a. [The foregoing method of comput- 

 ing term by term, may of course be employed for developing any case 

 of A n (x ± 0)"]. 



(26) The remainder of the present communication will be quite in- 

 dependent of the rule of Newton, and of everything that has preceded, 

 except the group of Criteria at page 344. These formulae [3] are 

 merely deductions from the principle of De Gua ; but as we shall have 

 frequent occasion to advert to the numerical multipliers connected with 

 the formulae [3], and as these same multipliers are those employed in 

 the rule of Newton, we shall for brevity and convenience refer to them 

 under the denomination of Newtonian factors. The following general 

 property will be found useful in the business of actual solution. 



If an equation be represented by the notation f(x) - 0, and its roots 

 be each diminished by r, the transformed equation will be f(x + r) = 0, 

 each of the roots (x) of which will be less by r than the corresponding 

 root (x) of the original equation :* and we know from the theory of 

 equations, that f(x + r) may be written either 



f(x + r) =/(*) +Mx)r + \flxy + ^{xY 



1 



2.3.4 . . . ft 



-f n {x)r n = 0 



or 



1 



2.3.4 . . . f£ 



fn(r)x» = 0 



►[1] 



Now the property we propose to prove is this, namely :— If the middle 

 one of any three of the consecutive functions, 



m, /.(*)> \fj,*), ~/,^), • • . . [2] 



* Of course, although we here speak of the roots being " diminished" by r, it will be 

 understood that r may be regarded as either positive or negative. Indeed the property 

 in the text holds whether r be real or imaginary, as the general demonstration of it proves. 



