VOL. XXXIV.] PHILOSOPHICAL TRANSACTIONS. 147 



lab is greater than 4ah ; therefore a -\- h y. x^, the square of the second term, 

 will be greater than Aab X x" the quadruple product of the first and third 

 terms. 



Pkop. II. — In any cubic equation, all whose roots are real, the square of the 

 second term is always greater than the triple product of the first and third. 



If the cubic equation has all its roots real, they may be represented with their 

 signs by a, b, c, and the equation will be expressed thus : 

 y^ — ny'^ + aby — abc = O. 



— c/ + bey 



But by lemma 2, a- + b' + c^ is always greater than ab -\- ac -\- be ; and 

 consequently, adding 2ab + 2crc + 2bc to both sides, d^ + b'- + c'- + 2nb + 

 2ac + 2be ( = a -\- b + c) will be greater than Sab + 3ac + 3 be; and 

 therefore a -\- b + c' X y" must be greater than 3cib + Sac + 3bc X y\ that 

 is, the square of the second term must be greater than the triple product of the 

 first and third terms. 



Cor. 1. — In general, it appears from the demonstration, that the square of 

 the sum of three real quantities, « + i + c is always greater than the triple 

 sum of all the products that can be made by multiplying any two of them into 

 each other. 



Cor. 2. — It follows from the proposition, that when the square of the second 

 term is not greater than the triple product of the first and third terms, the 

 roots of the equation cannot be all real ; but two of them must be impossible: 

 and this plainly coincides with one part of Sir Isaac Newton's rule for discover- 

 ing when the roots of cubic equations are impossible. He desires we may write 

 above the middle terms of the equation the fractions 4, J- J- 



-J- as in the margin; and placing the sign + under x^ -\- px" -\- (jx -\- r = O 

 the first and last term, he multiplies the square of the + — ^jc + 

 second term by the fraction i that is above it ; and if the product is greater 

 than the product of the adjacent terms, he places -f- under the second term ; 

 but if that product is less, he places — under the second term, and says, there 

 are as many impossible roots as changes in the signs. Now by this proposition, 

 if p- x* is not greater than 3c/x^, or 4- j/ x"' greater than gx*, the roots cannot 

 be all real. The same supposition makes two changes in the signs, whatever 

 sign we place under the third term, since the signs under the first and last are 

 both + ; and therefore this proposition demonstrates the first part of Sir Isaac 

 Newton's rule, as far as it relates to cubic equations. 



u '2 



