348 



all the triads of which satisfy the conditions [3],* namely, 



1st. 2«x4x8 > (ra- 1)9 2 , that is, 256 > 243 

 2nd. 3(n - 1)9 x 4 > 2(n - 2)8 2 , „ 324 > 256 

 3rd. 4(^-2)8 x 8 > 3(« - 3)4 3 , „ 512 > 48 



and yet, only two of the four roots are imaginary ; a real root lies be- 

 tween 0 and J. For diminishing each root by -5, we have 



1-2-25 +2 -1 + -2(-5 



•5 - -875 -5625 - -21875 



- 1-75 + 1-125 - -4375 - -01875 



the change of sign showing that a root lies between 0 and *5. We shall 

 find, upon trial, that the first figure of this root is -4. 



This example clearly enough shows that the fulfilment of the condi- 

 tions [3], one after another, without interruption, by the successive 

 triads of an equation, is no proof that more than a single pair of imagi- 

 nary roots enter ; one pair there necessarily must be ; other pairs there 

 may be, but there are not other pairs necessarily. If the absolute 

 number in the foregoing equation had been 1*4, or any greater number, 

 instead of the number -8, all the roots would have been imaginary, 

 although only two imaginary roots could have been indicated by the 

 preceding tests ; for the coefficients would then have satisfied the con- 

 dition 



(4A,A 2 - A 3 *) A 0 > A,A\, 



which has place only when all the roots of the equation are imaginary, 

 as may be proved as follows : — 



(11) Let the general equation of the fourth degree, with the last 

 term positive, be put in the form 



x 2 (A i x 2 + A 3 x + A 2 -p) f (px^ + AtX + A 0 ) = 0. 



Then, if p be taken equal to A? 4- 4^4 0 , the second of these qua- 

 dratic expressions, being a square, will be positive (or zero) for every 

 real value of x ; and the first will be always positive also, that is, all 

 the roots of the equation will be imaginary, if 



* If the fourth term were 4- 4#, instead of - 4x, the first and last triads would each 

 still satisfy the conditions ; but the middle triad would fail. Yet, as the imaginary roots 

 indicated by the first triad are indi«ated in the positive region of the roots, and the ima- 

 ginary roots indicated in the last triad would then be indicated in the negative region, 

 we should know that the two pairs of roots are distinct. As it is, however, all the roots 

 are indicated in the positive region. 



