p(x) = qo(x) * Tn +1 (x) + ro(x) YTn~l(x =-q1(x)W ro(x) + r1(x) ro(x) = q2-(X) * ri(x) + r2(x) r1() = q3(x) *r2(x) + r3(X), etc., where the (degrees of thie ri formr a strictly decreasing sequence. From these equations we may write r,(Q) = i(x) -.p(.x) + bi(x) . T.+1(x), where ai and bi are defined recursively by (a,=ai-. -qiai-?, a-1=O, a2= 1 lbi = bi-) - lti, b-, ) b1 = 1, b.2 = O. It may be proven that the sutm of the degrees of a,(x) and ri(x) is at most n. The first set of equationis may be written p(x) -. [ri(x)/a(x))] - [b(x)/ai(x)1*T.+1(x), so that ri(x)/a,(x) is a rationIal approximation to p(x), exact wherever Tn+1(x) vanishes. Since T,,(.r) I I inl the interval of approximation, bi(x)/ai(x) provides