equations of all orders , by continuous approximation. 311 
scale of known quantities, therefore, by which we advance so 
rapidly in the first process, fails in those which follow. 
7. Still we can reduce these formulae to known terms ; for 
since we have in general 
D r D J (px = — j — - \ —7- D ^ <poc 
(See Arbogast, § 137); by applying a similar reduction to 
the successive terms in the developement of D w pR' = D” 
(R + r )> we obtain* 
D”ipR'=D”$R + ^ D’"+ I <jR . r+ ^f- 1 . D’"+ 2 <jR .f 
+ m+ p D^+^R . r 3 + &c. 
And it is manifest that this expression may be reduced to a 
form somewhat more simple, and at the same time be ac- 
commodated to our principle of successive derivation, by 
introducing the letters A, B, C, &c. instead of the functional 
expressions. 
8. As a general example, let 
M = D m $>R + Nr 
N =D m+1 <pR + Pr 
P =D m+2 <pR + Qr 
* This theorem, of which that in Art. 4 is a particular case [m — 0], has been 
long in use under a more or less restricted enunciation, in aid of the transformation 
of equations. Halley’s Speculum Analyticum, Newton’s limiting equations, and 
the formulae in Simpson’s Algebra (ed. 5, p. 166, circa Jin.) are instances. In a 
form still more circumscribed [mi, R=o, 1, 2, &c.] it constitutes the Nouvellc Me - 
tbode of Budan ; which has been deservedly characterized by Lagrange as simple 
and elegant. To a purpose which will be noticed hereafter, it applies very happily ; 
but regarded as an instrument of approximation, its extremely slow operation renders 
it perfectly nugatory : and as Legendre justly reported, and these remarks prove, 
it has not the merit of originality. 
