34 Proceedings of the Royal Irish Academy. 



(8.) If we agree to call that particular root of Q = 0, which lies in 

 the interval [ri, rn], the indicator of the two imaginary roots of the 

 cubic equation -P = 0, we may infer, from what is shown above, that 

 when the two roots of the indicating quadratic are real, if they be 

 positive (they must always both have the same sign), the indicator is 

 a positive number ; and if they are negative, the indicator is a negative 

 number ; that is to say, the region in which the real roots of the indi- 

 cating quadratic lie is the region in which the indicator itself lies. 

 And we have seen that that root of the derived quadratic equation 

 Q = 0, which is the indicator, is the only one of the two roots of that 

 equation which lies between the roots r^, ?'3 of the equation Q^ - 2>PP' 

 = 0, the other root of Q = being excluded from that interval. 



The real root, however, of the proposed cubic equation P = must 

 lie outside the interval [r^, r2] ; for, as shown above, every value of a:, 

 withiyi this intevval, causes the expression Q--3PP' to be negative; 

 whereas the value of x, which satisfies the condition P=0, reduces 

 that expression to Q^, which is positive ; and, for a similar reason, the 

 root of the simple equation P' -Q must lie without the interval. 



That the real root of the cubic equation lies icithout the inteiwal 

 \j\, ro] is a conclusion that might have been immediately deduced from 

 the foregoing truth ; namely, that that root of Q = 0, which is not the 

 indicator, does itself lie without the interval. For, since this root 

 separates the real root of the cubic fi'om the imaginary pair, or rather 

 from the indicator oi that pair, the seal root of P = must occupy a 

 place more remote from the interval [j'l, r^ than is the place of that 

 root of Q = 0, which is not the indicator of the imaginary pair. 



(9.) It thus appears that, without any preliminary analysis of the 

 cubic equation P = 0, we can always ascertain, from an examination of 

 the quadratic equation Q^-2>PP' =0 — 



First, whether the equation P = has a pair of imaginary roots or 

 not. 



Second ; if it have imaginary roots, lohicJi of the two roots of the 

 derived quadratic, (3=0, it is that is the indicator of the pair ; it is 

 that one which lies between the two roots of Q- - 3PP' = ; and only 

 one of the two can so lie. 



And, thirdly, we learn that the real root of the cubic always lies 

 outside the interval between the two roots of the equation last men- 

 tioned; and from thus knowing the interval from which it is excluded, 

 the first figure of it becomes the more readily determinable, whenever 

 all three of the roots are indicated, by the signs of the terms of the 

 equation, in one and the same region. 



"We may fiu'ther observe here, that whenever we seek to deter- 

 mine the character of a pair of dotibtful roots in a cubic equation 

 P = (and which roots are indicated in an interval comprehending 

 also a real root of Q = 0), by the process of continuous approximation, 

 we may be sure, if the roots be imaginary, that the indication of ima- 

 ginarity will not be arrived at till the approximating number reaches 

 one or other of the values r^, or n; after arriving at which, the condi- 



