124 Prof. Young on the Evaluation of Vanishing Fractions, 



which it could not do unless the three corresponding coefficients 

 of the primitive fulfilled that condition : hence in this case also 

 the roots of the primitive must all be equal, or else two at least 

 must be imaginary. The general inference, then, is this : which- 

 ever triad furnishes the condition of equality, the equation must 

 have either two imaginary roots at least, or else all its roots must 

 be equal. When, therefore, the equation is not a complete 

 power, the sign = , in the application of Newton's criteria to the 

 several triads of coefficients, is to be regarded as significant of the 

 same thing as the sign >. 



It may be observed here that we know that the equation 

 ?w(# — a) n =0 will have all its roots equal to a; and that by de- 

 veloping the first member by the binomial theorem, the above- 

 mentioned condition of equality will have place throughout; 

 and further, that when this condition has not place throughout 

 all the triads, the first member of the equation cannot be the de- 

 velopment of m(x— a) n , — in other words, that the roots of the 

 equation cannot be all equal. Hence the theorem, that, if the 

 roots are all equal, the condition of equality has place for every 

 triad, holds conversely ; that is, if the condition has place for 

 every triad, the roots are all equal. 



In no subject is greater caution needful, in dealing with the 

 converse of established theorems, than in the Theory of Equa- 

 tions ; for the most part they are inconvertible. It has been 

 regarded as questionable, or " more than questionable," whether 

 the converse of Newton's rule is true ; but there ought to be no 

 doubt at all on the matter. Equations, in any number, may be 

 proposed, for which not one of Newton's criteria of imaginary 

 roots holds, and yet of which the roots shall be all imaginary. 

 To be convinced of this, it is sufficient to refer to the theorem 

 marked (II.) at the end of my paper in the Journal for October 

 1865, to consider A 3 as negative, and A T as a comparatively 

 small coefficient. Erom the theorem marked (I.), too, it is at 

 once seen that if p in the cubic be negative, two roots of it may 

 be imaginary, though Newton's rule could never make known the 

 fact. And Newton himself was fully aware of the inconvertibility 

 of this rule ; for, when speaking of the marks by which to dis- 

 tinguish the entrance of positive roots from the entrance of ne- 

 gative roots, he says (I still refer to Raphson's translation), 

 " And this is so where there are not more impossible roots than 

 what are discovered by the rule preceding. For there may be 

 more, although it seldom happens." 



Note. — In the opening paragraph of my paper in the Journal 

 for May last, the word " holds " should be replaced by "holds 

 or fails :" the next following paragraph, however, sufficiently 

 implies that this was intended. 



