34 On certain Criteria of Imaginary Roots of Equations. 



ing one in the former set must hold also, yet one of these may 

 fail without any failure in the corresponding form of the pre- 

 ceding set. 



The latter forms, when expressed in words, furnish the 

 following rule, which it will be easy to remember: — 



Commencing with the second coefficient and proceeding 

 towards the right, or with the last but one and proceeding 

 towards the left, multiply the successive coefficients by any 

 series of consecutive whole numbers, and square the results : 

 the square of each coefficient will thus be multiplied by an 

 integral square. Let the product of the adjacent coefficients, 

 on each side, be now multiplied by the same square minus 1 ; 

 and compare these results with the former, as in Newton's 

 rule. 



Of course criteria of this kind, which are to be applied im- 

 mediately to the coefficients of the given equation, cannot 

 always be expected to make known the exact number of ima- 

 ginary roots entering that equation ; yet our chances of ob- 

 taining this knowledge become multiplied with the number of 

 our distinct and independent tests for the detection of such 

 roots. At present we have no easily applied, and at the same 

 time completely decisive tests of this kind which extend be- 

 yond an equation of the third degree; yet, as respects equa- 

 tions of the fourth degree, I think a little may be added in 

 this way, as inferences from the following general expression 

 for two roots of the biquadratic equation — 



x 4 + p a? + q x + r = 0, 



when the other two x v x% are given : — 



The inferences adverted to are these, viz. — 

 Supposing q positive, which is always allowable, we see 

 from this expression that,— 



1. If two real roots occur in the positive region, the other 

 two roots must be real also. 



2. If two imaginary roots are indicated in the negative re- 

 gion, the others must be imaginary. Therefore, 



3. If two real roots are detected, without regard to their 

 situation, and the remaining two are indicated in the negative 

 region, these must be real also; and if two imaginary roots 

 are indicated in either region, and the remaining two are in- 

 dicated in the positive region, these must be imaginary also. 



Belfast, June 11,1846. 



* Analysis, &c. of Cubic and Biquadratic Equations, p. 235. 



