367 



and from the principle above, we readily see, withont any calculation 

 with these large numbers, that since 



13 - 64 + 62- ... is positive, 



the remaining roots of the cubic must be imaginary. 



Bat it may be well to show here the amount of numerical labour 

 spared when the character of the roots, from close proximity to equality, 

 is much less readily discoverable, by the ordinary method, than in this 

 example. 



By the method here proposed, the work is 



18 -64 = -46 



62-46326147 

 16-46326147 



By the common rule (after multiplying the absolute number by 4), 



249-85304588 

 (15-2771) 2 = 233-38978441 



16-46326147 



If the first of the expressions [6] be itself positive, we should know, 

 at the outset, that two roots are imaginary. If it be zero, then, since, 

 when increased by a positive quantity, the result would be positive, 

 we should know then, also, at the outset, that two roots would be 

 imaginary ; and similarly in reference to the second triad of coefficients, 

 as the coefficients may be reversed. But both of these conclusions may 

 be inferred from, what has been previously established. 



(28) Returning now to the first of the equations, p. 364, if we re- 

 present the original polynomial f(x) by X 0 , and the successive co- 

 efficients of r, r 2 , &c, in [1], that is, the several derived polynomials, 

 by Xi, X 2 , &c, respectively, a very general form may be given to the 

 criteria of imaginarity at p. 344, namely, 



2nX 0 X 2 > (n-l)X* 

 3(n - 1)^X3 > 2(» - 2)X 2 2 

 4(n - 2)X 2 X 4 > B(n - 3)X 3 * 

 5(n - 3)X 3 X 5 > 4(n - 4)X 4 2 



2nX n _ 2 X n > (n - l)XVi* 



in which expressions the x involved in X 0 , X u &c, may take any real 

 value, positive or negative, whatever. If we put x = 0, the formulae 

 become those at page 344 ; X 0 , Xj, &c, and A 0 , A lf &c, then being 

 identical. Take, for example, the equation 



* X n is, of course, always identical with A n . 

 He T. A. PEOC. VOL, X. 3D 



