148 PHILOSOPHICAL TRANSACTIONS. [aNNO 1720. 



Cor. 3. — If the second term is wanting in a cubic equation, and the third is 

 positive, two of the roots of the equation must be impossible : for the square 

 of the second term (equal to nothing in this case) will be less than the triple 

 product of the adjacent terms. But this will better appear from considering 

 that, when the second term vanishes in an equation, the positive and negative 

 roots are equal, and when added together, destroy each other. Suppose the 

 roots to be + a and — b, — c; tlien in this case a = -\- b -\- c, and the co- 

 efficient of the third term will be — ab — ac -\- be =■ — A' — ibc — c'- -|- 

 bc z=i — b"^ — be — c^, and consequently negative. Or, if we suppose two 

 roots positive and one negative, let them be — a, + b, + c, then the coeffici- 

 ent of the third term will be still — b"^ — be — c'-. Therefore when the roots 

 are real, the coefficient of the third term is negative ; and if the coefficient of 

 the third term is not aifected with a negative sign, it is a proof that two of 

 the roots are impossible. 



Prop. 3. In any cubic equation, all whose roots are real, the square of the 

 third term is greater than the triple product of the second and fourth terms. 



In the same cubic equation, whose roots are a, b, c, the square of the third 

 term is ab -f- ac -\- be , the product of the second and fourth terms is a'bc 

 -\- ab'^c -\- abc\ as is plain from the inspection of the equation ; and it is 

 obvious that a"bc -f ab'^c -j- abc^ is the sum of the products of any two 

 of the terms ab, ae, be; and therefore, by Corol. 1, Prop. 2, the square of 

 the sum of these terms, that is, ab + ac + be , must be greater than 3d'bc -{- 

 ?,ab'^c -}- 3ac-b. So that ab -\- ac -\- be X y' must be greater than 

 3d^bc -|- Zab'^c -j- 3ac^b X y"^; that is, the square of the third term must be 

 greater than the triple product of the second and fourth terms. 



Cor. 1. It follows from the demonstration, that ab -\- ae -\- be is always 

 greater than 3abe X a -^ b -\- c. 



Cor. 2. If the square of the third term is found to be less than the triple 



product of the second and fourth terms, then the roots of the equation cannot 



be all real quantities; and this agrees with the second part of Sir Isaac Newton's 



rule for finding when the roots of a cubic equation are 



i ' 



impossible : for this case gives — to a + px^ -f- ^.i^ -|- r =: O 



be placed under the third term, and -|- :^ — -f- 



consequently two changes of the signs, whatever sign is placed under the 



second term. 



Sehol. After the same manner it may be demonstrated, that in a cubic equa- 

 tion, whose roots are all real, if the second term is wanting, the cube of the 

 third part of the third term taken positively, is always greater than the square 



