equations of all orders, by continuous approximation. 319 
seem more naturally to depend on the simple derivees, a , b , &c. 
than on A, B, &c. their aggregates ; and since, in fact, as long 
as ris assumptive orindependent of R, our system of derivation 
offers no peculiar advantage ; I should prefer applying the 
limiting formulas in the usual way ; passing however from 
column to column (Wood, § 318.) of the results, at first by 
means of the neat algorithm suggested in the note on Art. 7, 
and afterwards by differencing, &c. as recommended by La- 
grange, (Res. des Eq. Num. § 13), when the number of co- 
lumns has exceeded the dimensions of the equation. (Vide 
Addendum.) 
If, during this process the observation of De Gua be kept 
in view, that whenever all the roots of <px are real, D™” 1 <px 
and D m+I <px will have contrary signs when D m <px is made 
to vanish, we shall seldom be under the necessity of resorting 
to more recondite criteria of impossibility. Every column in 
which 0 appears between results affected with like signs, will 
apprize us of a distinct pair of imaginary roots ; and even a 
horizontal change of signs, occurring between two horizontal 
permanences of an identical sign, will induce a suspicion, 
which it will in general be easy, in regard of the existing 
case, either to confirm or to overthrow. 
as. The facilities here brought into a focus, constitute, I 
believe, a perfectly novel combination ; and which, on that 
account, as well as on account of its natural affinity to our own 
principles, and still more on account of the extreme degree of 
simplicity it confers on the practical investigation of limits, 
appears to merit the illustration of one or two familiar ex- 
amples. 
