310 Mr. Horner’s new method of solving numerical 
Where by D” <pR is to be understood ^ , viz. the n ,b 
derivee with its proper denominator ; or, that function which 
Arbogast calls the derivee divisee, and distinguishes by a c 
subscribed. Having no occasion to refer to any other form 
of the derivative functions, I drop the distinctive symbol for 
the sake of convenience. Occasionally these derivees will 
be represented by a, b, c, &c. 
5. Supposing <pR and its derivees to be known, the mode 
of valuing <pR' or <p(R-|-r) is obvious. We have only to 
say in the manner of Lagrange, when preparing to develope 
his Theory of Functions, 
<pR' = <pR -|- Ar 
A D<pR -J- Br 
B — D‘(fiR Cr 
C = D 3 <pR + Dr 
V = D— 4 <pR + Ur 
U=D"->R + r CIO 
Taking these operations in reverse order, we ascend with 
rapidity to the value of <p (R + r) or <pR'. 
6. The next point is, to apply a similar principle to dis- 
cover the value of (p (R + r -f- r r ) = <p (R' + r') = <pR". We 
here have 
<pR ,/ = cpR / + A'r' 
A' = DpR' + B' r' 
B' = D>R'+ C' r' 
C =D 3 ?R'+D 'r' 
V' ^D^R'+U'r' 
U' = D W_I pR'-f- r # 
But the former operation determined <pR' only, without giving 
the value of any of the derived functions. The very simple 
