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Prof. Sylvester on Newton's Rule for 



[April 7, 



for assigning an inferior limit to their number. He has given no 'clue 

 towards the ascertainment of the grounds upon which this rule is based, and 

 has stated it in such terms as to leave it quite an open question whether or 

 not he had obtained a demonstration of it. Maclaurin, Campbell, and 

 others have made attempts at supplying a demonstration, but their efforts, 

 so far as regards the more important part of the rule, that namely by 

 which the limit to the number of imaginary roots is fixed, have completely 

 failed in their object. Thus hitherto any opinion as to the truth of the rule 

 rests on the purely empirical ground of its being found to lead to correct 

 results in particular arithmetical instances. Persuaded of the insufficiency 

 of such a mode of verification, the author has applied himself to obtain- 

 ing a rigorous demonstration of the rule for equations of specified degrees. 

 For the second degree no demonstration is necessary. For cubic equa- 

 tions a proof is found without difficulty. For biquadratic equations the 

 author proceeds as follows. He supposes the equation to be expressed 

 homogeneously in .r, y, and then, instituting a series of infinitesimal linear 

 transformations obtained by writing x-\-hy for a?, ox y-\-hx for y, where h is 

 an infinitesimal quantity, shows that the truth of Newton's rule for this 

 case depends on its being capable of being shown that the discriminant of 

 the function (1, +e, +e, 1 3/)* is necessarily positive for all values of 

 e greater than unity, which is easily proved. He then proceeds to consider 

 the case of equations of the 5th degree, and, following a similar process, 

 arrives at the conclusion that the truth of the rule depends on its being 

 capable of being shown that the discriminant, say (D) of the function 

 (1, e, rfi t), l^jo, y^f , which for facility of reference may be termed 

 " the (e, 77) function," is necessarily positive when e^—er\^ and rj^—rje^ are 

 both positive. This discriminant is of the 12th degree in e, rj. But on 

 writing cc=€r], y=e^ + r]\ it becomes a rational integral function of the 6th 

 degree in x, and of the second degree in y, and such that, on making 

 D = 0, the equation represents a sextic curve, of which x, y are the 

 abscissa and ordinate, which will consist of a single close. It is then 

 easily demonstrated that all values of e, rj which cause the variable point 

 Xy y to lie inside this curve, will cause D to be negative (in which case the 

 function e, rj has only two imaginary factors), and that such values as cause 

 the variable point to lie outside the curve, will make D positive, in which 

 case the e, 77 function has four imaginary factors. When the conditions 

 concerning e, 1/ above stated are verified, it is proved that the variable 

 point must be exterior to the curve, and thus the theorem is demonstrated 

 for equations of the 5th degree. 



The question here naturally arises as to the significance of the sign of D 

 when such a position is assigned to the variable point as gives rise to imagi- 

 nary values of e, 77, which in such case will be conjugate quantities of the 

 form \ + X—^/>t respectively. 



The curve D will be divided by another sextic curve into two portions, 

 for one of which the couple e, 77 corresponding to any point in its interior is 



