312 Mr. Horner’s new method of solving numerical 
represent any successive steps in the series in Art. 5; then are 
D* <pR = M — Nr 
D m+I <pR = N - Pr 
D^+^R = P — Qr 
And by substituting these equivalents in the developement 
just enounced, it becomes 
D m <pR'= M -f m Nr -f P r 1 + m ~ m \ Z Qr 3 + &c. 
9. With this advantage, we may now return to the process 
of Art. 6 , which becomes 
<pR"= <pR 7 + A V 
A'= ( A + Br + C r 8 + Dr 3 + Er 4 + &c.) -f B' r' 
B' = (B + 2O + 3 Dr* + 4 Er 3 + &c.) + CV 
C 7 = (C + 3 Dr+ 6 Er- + &c.) + D 7 r' 
V'=(V + ^Ur+ n -~ z - : ~Y V ) + U 7 r' 
U 7 ==(U + ^r) +r' .... CII-1 
Taking these operations in reverse order as before, by deter- 
mining U / , V' .... C', B', A 7 , we ascend to the value of <pR". 
10. In this theorem, the principle of successive derivation 
already discovers all its efficacy ; for it is obvious that the 
next functions U' 7 , V" C ", B", A", <pR'", flow from the 
substitution of A', B', C 7 , . . . . V', U 7 , <pR", r', r" , for A, B, 
C . . . . V, U, <pR', r, r', in these formulae ; and from these 
U 7/7 , V /7/ , &c. ; and so on to any desirable extent. In this re- 
spect, Theorem II, algebraically considered, perfectly answers 
the end proposed in Art. 2. 
11. We perceive also, that some advance has been made 
toward arithmetical facility; for all the figurate coefficients 
