344 



[2] to the last three coefficients of each, we shall arrive at the series of 

 conditions which here follow ; and the existence of any one of these 

 conditions will imply the existence of at least one pair of imaginary 

 roots in the primitive equation. 



Conditions of imaginary roots in [/]. 



2nA 0 A 2 > {n-l)A x 2 

 2>(n-\)A l A z >2(n - 2)A? 

 4.(n-2)A 2 A i >3(n-3)A 3 * ... [3] 

 5(n-S)A z A 5 >4(n-4:)A i 3 . 



2nA n . 2 A n > (w- l)-4Vi- 



(4) Prom a mere inspection of this group of conditions, it is obvious 

 that all are comprehended in the general formula 



(m + 1) (n - m - 1) A m .^A m ^ > m{n - m) A} m .... [4], 



where m is the exponent of x in the middle one of any three conse- 

 cutive terms of the equation [/]. And from this general formula we 

 at once see that the difference of the numerical multipliers 



(m + 1) (n - m - 1), and m(n - m) 



is always the same for the same value of n, namely, n + 1 • "We may 

 therefore express the above conditions somewhat differently, thus : — 



[(» - 1) + (* + 1)] A 0 A 2 > (n - 1) A x * 

 [2{n - 2) + (n + 1)] A,A 3 > 2{n - 2) A* 

 [3(n - 3) + (ft + 1)] A 2 At> 3(n - 3) A 3 \ . . [5] 

 [4(w - 4) + (ft + 1)] A z A b > 4(ft - 4) A? 



[(n - 1) + (n+ 1)] A n .,A n > (n - 1) A n *_ x ; 



so that the multiplier for the square of the middle term of any triad of 

 terms, increased by the constant number n + 1, will always be the mul- 

 tiplier for the product of the extreme terms ; and therefore, iu applying 

 the tests, it will usually be the more convenient to deal with the 

 squares first. 



(5) It is evident that the foregoing inequalities, for an equation of 

 the wth degree, are n - 1 in number ; and from the general expression 

 [4] we see that the multiplier which enters the middle term of each of 

 the two completed series of terms, when n is even, or the number of 

 terms in each completed series odd, is always a square number : thus, 



