356 



2. 3x 5 + Sx l - bx z + \2x 2 - 7x + 9 = 0 

 + + + 



The last triad here need not be tested, as the row of signs already 

 written down terminates in a minus sign. One pair of imaginary 

 roots is detected in the positive region. 



3. 7a 4 - 2x z + 8x* - 5% + 17 = 0 

 + - + - 



In this equation all the roots are imaginary. 



In applying the tests it will always be found preferable to employ 

 them in the forms marked [5] at page 344. The multipliers in the 

 right-hand members of these forms are all included in the general ex- 

 pression m [n - m), and those in the left-hand members, are these each 

 increased by the constant number n + 1 . In an equation of a high 

 degree, the easiest way of proceeding will be to place under the second 

 term of the equation n- 1, under the third term 2 (n-2), under the 

 fourth term 3 (n- 3), and so on; in other words, commencing at the 

 second term, to multiply the exponents by 1, 2, 3, &c, placing the 

 results underneath; remembering that these numbers will recur in 

 reverse order when the middle term, or the first of the two middle 

 terms, is reached : each will be the multiplier for the square of the 

 term under which it is placed ; and when this multiplier is increased 

 by the constant number n + 1, that is, by the leading exponent plus 

 1, the result will be the multiplier for the product of the two extreme 

 terms of the triad we are testing : thus, 



4. 5x* - 'Ix 1 + 3x 6 - 24a 5 - 16a 4 + x 3 - \x* - 2x - 60 = 0 



7 12 15 16 15 12 7 



These numbers are the multipliers for the squares of the coefficients 

 immediately above them; and those for the product of the extreme 

 coefficients of the triad, found by adding 9 to each number, are — 



16 21 24 25 24 21 16 



or, expunging common factors, the two rows of numbers will be those 

 here underwritten : 



5x* - 2X 1 + Sx Q - 24%* - 16a 4 + x z - 4x 2 - 2x - 60 

 For the squares, 



7 4 5 16 5 4 7 



For the products, 



16 7 8 25 8 7 16 



And since 



16.5.3 < 7.2 2 , 7.2.24 > 4.3 2 , - 8.3.16 < 5.24 2 , - 25.24.1 < 16. 16 2 , 

 8.16.4 > 5.P, - 7.1.2 < 4.4*, and 16.4.60 > 7.2 2 , 

 the row of signs will be 



+ + _ + + _ + _ 



