116 Prof. Young on the Demonstration of Newton's Bute for 



a set of three terms for which the condition fails, thus putting 

 a stop to the series of concurring indications, and preparing the 

 way for new, distinct, and independent indications. 



[Now it is to be specially noticed that the last set of three 

 terms in any of the cubics is not in the slightest degree influ- 

 enced, as to their satisfying the criterion or not, by the first set 

 in that cubic : the two sets, as to this criterion, are wholly inde- 

 pendent; if both sets satisfy it, they do so quite independently 

 of each other. But the case is very different as respects the 

 last set in one cubic and the first in the cubic next following. 

 The condition is satisfied or not, in this set, accordingly as it is 

 satisfied or not in the preceding set. This distinction between 

 the two cases it is of importance to observe and attend to : the 

 last set in any cubic, as to satisfying or mot the condition of 

 imaginary roots, has nothing whatever to do with any preceding 

 set ; it is totally independent of and uninfluenced by all ante- 

 cedent results ; and if the last set in any cubic should indicate 

 imaginary roots as well as the first set in that cubic, everybody 

 knows that, in the process of developing the real root, the indi- 

 cation would disappear from the final terms, but would still be 

 retained in the leading terms. Yet it is the final set, and that 

 final set alone, that determines the character of the leading set 

 in the cubic next following ; so that whatever this character may 

 be, no feature of it can possibly have been transmitted to it, as 

 from an origin, by any antecedent leading set whatever. Having 

 interpolated these remarks, I now proceed to the third con- 

 clusion.] 



3. So soon as the criterion is again satisfied, the condition 

 being entirely independent of and non-concurring with the for- 

 mer, must imply another and distinct pair of imaginary roots in 

 the primitive equation, and so on, to the end of the series of cubics. 



The criterion which is here supposed to be applied to the terms, 

 taken three at a time, of the successive cubics, supplies, one after 

 another, the entire series of conditions given in the margin above, 

 as will be seen presently. But as the last three terms of any 

 cubic always transfer their character, as respects the fulfilment 

 or non-fulfilment of the condition, to the three leading terms of 

 the cubic next following, the repetitions may be omitted. Attend- 

 ing to this, and applying the proper criterion to each of the fore- 

 going cubics in succession, we have the following conditions for 

 imaginary roots : — namely, 



1st. 2 2 .3 2 rc(rc-l)(rc-2) 2 A A 2 >2 .3 2 (n-l) 2 (?i-2) 2 A 1 2 ; 



2nd. 2 2 .3 3 (7i-l)(rc-2)A 1 A 3 > 2 3 .3 2 (rc-l) 2 A 2 2 ; 



3rd. 2 3 .3 3 .4(rc-2)(7i-3)A 2 A 4 > 2 3 . 3>-3) 2 A 3 2 ; 



4th. 2 3 . 3 4 . 4 . 5(7z-3)(rc-4)A 3 A 5 > 2 3 . 3 4 . 4 2 (rc-4) 2 A 4 2 ; 



