VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. GO I 



3d. In the Miscell. Anal, an equation is transformed into another, whose roots 

 are the squares, &c. of the roots of a given equation; and it is asserted, that there 

 are at least so many impossible roots -contained in the given equation, as there are 

 continual progresses in the resulting equation from + to -f and — to — . It is 

 afterwards remarked, that these rules sometimes find impossible roots when Sir 

 Isaac Newton's, and such like rules, fail ; and that Sir Isaac Newton's, &c. will 

 find them, when this rule fails. This rule may somewhat further be promoted by 

 first changing the given equation, whose root is x, into another whose root is 

 x>/ — 1 ; but, in my opinion, the rule of Harriot's, which only finds whether 

 there are impossible roots contained in a cubic equation or not, is to be preferred 

 to these rules, which, in equations of any dimensions, of which the impossible 

 roots cannot generally be found from the rules, seldom find the true number. 



4th. It is remarked, that rules which discover the true number of impossible 

 roots require immense calculations, since they must necessarily find, when the 

 roots become equal. In order to this, in the Miscell. Anal, there is found an 

 equation, whose roots are the reciprocals of the differences of any two roots of 

 the given equation ; and from finding a quantity (ir) greater than the greatest 

 root of the given, and (-) greater than the greatest root of the resulting equa- 

 tion, and substituting ir s -k — a, -k — 2a, &c. for x in the given equation ; will 

 always be found the true number of impossible roots. 



5th. In the same book are assumed 2 equations {nx"~ l — (n — \)px»- z -f- 

 ( n _ 2) qx"~i — &c. = O and x" — px"~ l -f- &c. = w), and thence deduced an 

 equation whose root is w, from which, in some cases, can be found the number 

 of impossible roots. 



6. In the "Miscell. Anal, is given the law of a series, and its demonstration, 

 which finds the sum of the powers of the roots of a given equation from its co- 

 efficients. Mr. Euler has since published the same in the Petersburg Acts. 

 Mr. De La Grange printed a property of this series, also printed by me about the 

 same time; viz. that if the series was continued in infinitum, the powers would 

 observe the same law as the roots, which indeed immediately follows from the 

 series itself; but from thence with the greatest sagacity he deduces the law of the 

 reversion of the series, y = a + bx + ex 2 -f- dx 3 -\- &c. : it has since been given 

 in a different manner from similar principles in the Medit. Analyt. 



7. In the Miscell. Analyt. the law of a series is given for finding the sum of all 

 quantities of this kind, a'" X (3" X / X S ! X &c. + &c. where «, |3, y, S, &c. 

 denote the roots of a given equation, from the powers of the roots of the given 

 equation. This law, with a different notation, has been since published in the 

 Paris Acts by Mr. Vandermonde; who indeed mentions that he had heard, that 

 a series for that purpose was contained in my book, but had not seen it. In the 



VOL. XV. 4 H 



